24-April-2017: LAB 15: PHYS 4A LAB: Collisions in two dimensions

Eric Chong

LAB 15: PHYS 4A LAB: Collisions in two dimensions

Lab Partners: Lynel Ornedo, Nina Song, and Joel Cook

Purpose: Look at a two-dimensional collision and determine if momentum and energy are conserved.

• Steel ball with steel ball
• Steel ball with glass ball

Introduction/Theory: Elastic collisions are collisions that conserve momentum and kinetic energy. This idea will be explored in this lab, as we collide two balls, in one case both made up of steel, and in another case one is steel and the other is glass. We will also trace the path of the balls after the collision. We do this because we know that the collision will conserve momentum in the x and y coordinates, and if the collision strays the two balls at certain angles, we can calculate the momentum afterwards to confirm if the momentum before and after are equal. In simpler terms, if one ball is rolling along the x-axis, and the other ball is at rest, that means that the initial momentum has no momentum along the y-axis. If the collision happens, then it is likely that there is momentum along the y-axis for both balls, but when added together, the total momentum should still be zero. This is the concept of conservation of momentum. We will also be looking at the momentum from the center of mass frame of reference to compare how it differs when we look at it from the lab frame of reference.

Procedure: We first setup the apparatus by clamping our phone with a camera on top of the apparatus. The apparatus is made up of a glass surface, a pole that extended high above the surface with a downward facing clamp for our phone. We also leveled the table, so that the ball will roll without speeding up or slowing down. Here is what it looks like:

Steel Ball with Steel Ball

We first used the balls made of steel and recorded their collision. On LoggerPro, we plotted their trajectories and graphed them on a position vs. time graph. We do not have a picture of the trajectory of the two balls during their motion, but we can show the graphs of their positions. What these graphs tells us is not only their positions but also their velocities, by looking at the slope of the lines. Here are the graphs:

The X and Y values are the coordinates for the moving ball, and the X 2 and Y 2 values are the coordinates for the ball at rest. From the graph we can see the velocities of the x and y coordinates for both balls by looking at the slope before and after the collision.

Before the collision:

After the collision:

We can now determine whether the momentum of the system is conserved after the collision. We do this by looking at the velocity of the x and y coordinates and calculating the momentum in the x and y direction separately. We measured the mass of the balls and got 67.0 g for the steel balls. We can calculate the final momentum of the balls by the summation of its masses multiplied with the final velocities, and then subtract with the summation of its masses multiplied with the initial velocities. Here is the calculation:

(Notice: the initial x and y velocities of the stationary ball is not completely zero, according to the graphs. This may be due to our imprecise plotting of the motion of the balls, or LoggerPro's estimate of the position of the ball while stationary.)

We also wanted to confirm if the kinetic energy is conserved. We converted the x and y velocities to the overall velocity by doing Pythagorean theorem, since kinetic energy is dependent on the overall velocity of the ball, not the x and y velocities separately. The calculation follows, the summation of 1/2 times their respective masses times their final velocities squared, minus the summation of 1/2 times their respective masses times their initial velocities squared. Here is the calculation for it:

As expected, the kinetic energy is conserved after the collision happened, thus confirming that this type of collision is elastic.

Afterwards, we wanted to compare our data to that of the center of mass. We made a separate graph and we then created a function that multiplied the x and y positions of both of the steel balls with their mass, all divided by the total mass, in order to get the position of the center of mass, for all of the data points. This is what the center of mass x and y graphs turned out:

X direction: Center of mass (yellow line)

Y direction: Center of mass (purple/pink line)

Upon observation, it seems that the x and y coordinates of the center of mass remains at a constant slope. In the x direction, the center of mass position is more so a constant straight line tilted slightly upward, and in the y direction the center of mass position resembles a downward line. What these graphs tells us is that the velocity of the center of mass does not change. In the center of mass reference frame, the collision is just a symmetric bounce, and the collision does not affect the center of mass at all.

We then tried to graph the x and y velocities of the center of mass by using the same function but instead multiplying the masses by velocity instead of position for each data point. We expected straight lines, but the result became this:

The data points became scattered, and this could be due to the limitations of LoggerPro and its data graphing, as it is difficult to plot with pinpoint accuracy throughout the course of the balls' motion.

Steel Ball with Glass Ball

For this next part, we replaced one of the steel balls with a glass ball and ran the same procedure as we did for the first half of the lab. Here are the results, with X and Y being the coordinates for the steel ball, and X 2 and Y 2 being the coordinates for the glass ball:

Before the collision:

After the collision:

We measured the glass ball, and its mass came out to be 19.8 grams. Here are the calculations for the conservation of momentum and kinetic energy:

(Note: there is a slightly significant error that appeared in the calculations for the y direction. The result is too high.)

After confirming that the momentum and kinetic energy are conserved, thereby making this collision elastic, we analyzed the center of mass. We did the same procedure as the first part, and here are the results:

Center of Mass X and Y Positions

Center of Mass X and Y Velocities

As expected for the center of mass positions, its slopes remained constant, thereby noting that the collision does not affect the center of mass. It is also expected that the velocity graphs of the center of mass is scattered, due to our imprecise plotting.

Conclusion: Some possible sources of errors for this lab may be from our imprecise plotting of the balls' movements, the table not being ideally leveled, and the collision losing a bit of energy in the form of vibrations, heat, and sound. Overall, the experiment showcases the concept of conservation of momentum and conservation of kinetic energy in elastic collisions. It also shows the concept that the collision does not affect the properties of the center of mass, for instance its velocity, and in the frame of reference of the center of mass, the two balls are bouncing off symmetrically.

19-April-2017: LAB 14: Physics 4A Impulse-Momentum activity

Eric Chong

LAB 14: Physics 4A Impulse-Momentum activity

Lab Partners: Lynel Ornedo, Nina Song, and Joel Cook

Purpose: Observe/verify the impulse-momentum theorem.

Introduction/Theory: The impulse-momentum theorem states that the change in momentum is equal to the net impulse. Impulse is also equal to the amount of force applied times the change in time. Using this idea, we can test it by measuring the area under the curve of a force vs. time graph for the collision of a cart. We can then measure the change in momentum after measuring the cart's mass and its velocity before and after the collision with a motion sensor. Through calculating both the area under the curve of the force vs. time graph and calculating the change in momentum through the change in velocity, we can verify the impulse-momentum theorem. Afterwards, we can also test what happens to the momentum when the collision is inelastic, or in other words, when the cart is stuck to the clay on a wooden post when it collides. We can use the same procedure and examine the impulse-momentum theorem for this case. In a collision, the force exerted on the cart is very large, since it essentially reverses the cart's motion, and is also very quick, possibly less than even half a second. Since this lab has a springy type of extension used for the collision, the maximum magnitude of the force would be when the extension is compressed the most.

Procedure: First we fastened the force probe on the cart so that a rubber stopper is extended in front of the cart. We then attached an upside down cart to a pole attached to the table so that the cart's springy extension could collide with the force probe's rubber stopper. The cart is on top of a relatively frictionless, level track, with a motion sensor at the end opposite the end with the upside down cart. Here is what the setup looks like:

Using LoggerPro, we used an experiment file to help setup the necessary graphs and sensor properties, and making sure that the push on the force probe is positive and that the velocity toward the motion detector is positive. Before we ran the test, we measured the cart's mass, which turns out to be 641 grams. We calibrated the force probe using the hook screwed into the force sensor and holding the sensor vertically with a certain mass attached to it and without it, and then zeroing the sensor after it is horizontal and attached to the cart.

Finally, we started the experiment by pushing the cart in order to get it moving and recording its velocity, and then observing the collision and measured the force with the force probe. We ended up with a graph that looks like this:

As expected for the initial appearance, it seems that the velocity changes from negative (away from the sensor) to positive (toward the sensor). The force also is very high and applied very quickly, as it was not even half a second. We then took the area underneath the force vs. time curve by taking the integral to measure the change impulse.

As seen, it seems that the impulse is 0.4132 N*s. We then looked at the velocity vs. time graph and observed the initial and final velocity. The measurements we used were -0.325 m/s, just before the collision, and 0.306 m/s, right after the collision. We then did the following calculation to calculate the change in momentum:

The result is relatively close to what we got for the area under the force vs. time graph. Since the calculated change in momentum of the cart is almost equal to the measured impulse applied to the cart, we have verified the impulse-momentum theorem.

For the next part, we added 500 grams to the cart, for a total of 1141 grams, and ran the same procedure. Here are the graphs:

The graphs are shaped similarly to those of the first experiment. The area under the force vs. time graph is 0.8575 N*s. For our calculated change in momentum, we did the same calculation but instead using the initial velocity as -0.424 m/s and the final velocity as 0.359 m/s:

The result, 0.893 N*s, is also relatively close with the value of 0.8575 N*s from the area under the curve. Because of this, we can conclude that the impulse and change in momentum are equal to each other, despite using a cart with more mass, thus verifying the impulse-momentum theorem for this case. Something to notice about these two experiments is that the final velocities do not have the exact same magnitude of the initial velocities, and this could be due to an imperfect spring that does not produce a perfectly elastic collision. The recoil can only produce a nearly elastic collision, and this is a limitation of our equipment, rather than the procedure.

For the last part of the experiment is the test in the case of an inelastic collision. For this experiment, we used clay and attached it to a wooden pole, instead of using the upside down cart with the springy extension. We also replaced the rubber stopper extension on the cart itself with a nail, so the cart will stick to the clay when it collides. Other than those changes, the procedure is the same, and the mass of the cart is the same as the one in the first experiment, with 641 grams (we should have used the same mass as the second experiment, as it says in the instructions, so this is an error on our part). These are the resulting graphs:

The appearance of the graphs are quite different than what we observed in the last two experiments. It is expected that the velocity will eventually reach zero, which it did. However, in the force vs. time graph, there seems to be a negative force enacted on the cart, which makes sense because the collision is inelastic, so when the cart is trying to rebound out of the clay, the clay makes sure the cart sticks to it, so the force has to pull the force sensor when the cart is in the rebound phase, thus making the value negative. The value we got for the area under the force vs. time graph is 0.3154 N*s. The impulse is smaller than the one in the nearly elastic collision, which is to be expected, since the cart does not reverse the direction of its velocity, and instead remains at rest after the collision, thus making the value lower. For our calculated change in momentum, we used the same calculations but instead using the initial velocity as -0.483 m/s and the final velocity as 0 m/s:

The result is relatively the same as the measured impulse. This is promising, as we can now conclude that the impulse-momentum theorem is verified for inelastic collisions as well. Through these experiments, we verified the impulse-momentum theorem for both nearly elastic and inelastic collision, as well as testing what happens when we change the mass of the cart.

Conclusion: Some uncertainties that exist in the experiment are from the springy extension from the upside down cart. The spring is not perfect, and this makes the collision not perfectly elastic. This is seen in our data, since the final velocities do not have the same magnitudes as those of the initial velocities. It is also possible that the friction in the track could have contributed to the slight decrease in magnitude of the final velocities.

Overall, the experiment was a success, since we are able to observe that the values for the change in momentum and the impulse are relatively similar. We can conclude that the impulse-momentum theorem is applicable to both inelastic and elastic collisions, and we can confirm this by taking the area under the curve of the force vs. time graph and calculating the change in momentum and comparing the results.

17-April-2017: LAB 13: PHYS 4A--Magnetic Potential Energy Lab

Eric Chong

LAB 13: PHYS 4A--Magnetic Potential Energy Lab

Lab Partners: Lynel Ornedo, Nina Song, and Joel Cook

Purpose: The goal of this lab is to verify that the conservation of energy applies to this system and to determine an appropriate function U(r) for the interaction between the magnets.

Introduction/Theory: The conservation of energy is a concept that says that energy is neither created nor destroyed. Energy is instead converted to other forms, such as gravitational potential energy, kinetic energy, spring potential energy, and magnetic potential energy. For the case of magnetic potential energy, there is no given equation that determines the amount of magnetic potential energy, unlike gravitational or elastic potential energy. Thus, given what we know, that for any system that has a non-constant potential energy, the potential energy is determined by the interaction of the force. By taking the integral of a force equation, we can find the equation for magnetic potential energy. Therefore, the challenge is to first find the force equation through observations and analysis of the relationship between the force of gravity and the magnetic force going against it. We can then take the integral of that equation from infinity to r, where r is the separation distance, and determine the equation of magnetic potential energy. Note that this equation will be applicable only to this type of magnet. Any other magnets would most likely have different equations. Finally, the last part of the lab is to verify that the conservation of energy is applied to this system. To do so, we can analyze the kinetic energy of the glider and observing its motion when a magnetic force is applied to the cart to reverse its motion.

Procedure: For the setup of the lab, we used a glider on an air track with a magnet attached to its end. Another magnet is also attached to one end of the air track, so the magnet on the glider will repel the magnet on the end of air track. A motion sensor is also attached to the same end of the air track for the second part of the lab. Here is what the setup looks like:





To determine the equation of U(r), we first measured the mass of the glider, and then raised the end of the track without the motion sensor and magnet to a certain height and measured the angle the track makes with the table. We also measured the distance the glider is from the magnet at the end of the track, since the magnets repel each other and the glider cannot enter a certain distance. We kept raising the track and measuring the angle and distance of the glider until we had 7 data points. Afterwards, we calculated the force of the magnet by setting the component of the gravitational force parallel to the track equal to the force of the magnet for all of the data points. Here are our data measurements:



We then plotted the force vs. distance graph, with the force being the magnetic force, and the distance being the distance between the glider and the magnet. Here is what the graph looks like:



We expected that the graph would have an equation of F=Ar^n, and as expected, we see that the data points created a power curve, with A being 1.864*10^-5 +/- 6.337*10^-6, and n being -2.421 +/- 0.08114. With this, we determined the magnetic force equation. Now we can determine the equation U(r) of the magnet by taking the integral from infinity to r. Here is the calculation:

Notice that the integral is negative, and this is because the force is going in the opposite direction of motion. Also, the infinity portion turns to zero because the exponent of little "a," when the numbers are plugged in, is negative, and this turns it to the denominator, and the limit when "a" goes to infinity is zero.

Now that we have the equation for U(r), we can verify that the conservation of energy. We first setup the glider at a reasonably close distance to the magnet. Then, we set up the motion detector to track the glider's movement. Then, we determined the relationship between the distance the motion detector reads and the separation distance between the magnets. From here, we now have a way to measure both the speed of the cart and the separation between the magnets at the same time. We then placed the glider at the other end of the track and made sure the motion sensor could detect it. Afterwards, we gave the glider a gentle push and recorded its movements on LoggerPro. On LoggerPro, we made a single graph showing Kinetic Energy, Potential Energy, and the total energy of the system as a function of time. We also included a graph with the position as the x-axis, instead of time, in order to see if there is a loss of energy as we gain distance. Here are what the graphs looks like:

As expected, the third graph showcases the magnetic potential energy rising as the glider gets close to the magnet, and then dropping once the glider starts moving away, and the kinetic energy dropping as it gets to the magnet, and then rising as it starts moving in the other direction, all while the total energy being relatively the same. However, from the second graph, we can see that there is a slight loss of total energy (the blue line), and this could be due to friction and air resistance.

Conclusion: From the experiment, there are a lot of uncertainties that could be found. For instance, the calculations to get the force equation is very delicate. Since we measured the distance the magnets are from each other in millimeters, a slight change in the numbers could make the uncertainty very high, and further make the rest of the calculations for U(r) to be off from what we expected. Also, in the second part of the experiment, we observed that we lost some of the total energy, and we mentioned that this was due to friction and air resistance. The friction could come from the uneven surface of the track, and the air resistance could come from the air that comes blowing from the track and onto the metal plate attached to the glider, and thereby reducing the glider's speed.

Overall, the experiment taught us not only the concept of magnetic potential energy, but also the steps to solve for the potential energy equations. We first observed what we could use to derive an equation, and in this case it is the component of gravitational force that is parallel to the track. We then took the integral from infinity to r, in order to find the equation of U(r). From here, we could verify that the conservation of energy is applied to the system. This experiment has a lot of practical value in many other experiments, and from our learning this procedure, the experiment was a success.

10-April-2017: Lab 11: PHYSICS 4A Work-Kinetic Energy Theorem Activity

Eric Chong

Lab 11: PHYSICS 4A Work-Kinetic Energy Theorem Activity

Lab Partners: Joel Cook and Max Zhang

Purpose: The goal of this activity is to observe the work done by a constant force, the work done by a non-constant spring force, and overall how the work-kinetic energy theorem works.

Introduction/Theory: The work-kinetic energy theorem states that work is equal to the change in kinetic energy. Through this definition, it is possible to calculate the work done on an object based on its change in kinetic energy. Energy is also transferred from one form to another. Energy cannot be destroyed nor created. Through this lab, we can highlight some of the mechanics that goes into the work-kinetic energy theorem in order to help us understand how energy in a system works. It is also noteworthy to mention that kinetic energy KE=0.5mv^2, where m is mass and v is velocity.

EXPT 1: Work done by a Constant force

Procedure: For this activity, we will be observing the work done by a constant force. We will be using a cart that is connected to a hanging mass by a string over a pulley on a leveled track. Here is a picture of the setup:

The first step is to set up a motion sensor at the back of the track in order to track the cart's distance. A force sensor with a string attached to the hanging mass is also set up on top of the cart. This way, the force and distance can be recorded simultaneously. We first zeroed the force sensor so it could accurately record the force of the hanging mass. We also made sure the track was leveled as best we could so the cart could move at constant speed. To make sure the cart doesn't fall off, we set up a bumper at the end of the track. After setting up Logger Pro and the sensors, we were ready for the next step. 

We used a 500 g mass and added it on top of the cart. We also used a 50 g hanging mass at the end of the string and pulled the cart back on the track. From here, we started to collect the data and let the car roll. Here are the graphs for the data:

As seen, the graph for the velocity vs. time is very unstable, and this could be due to the leveling of the track issues we faced. The force is constant throughout, which is expected, and is promising for our calculations.

From here, we made a kinetic energy axis on the force vs time graph. This allows us to compare the kinetic energy with the force vs time graph. Here is what it looks like (the graph at the very bottom):

As expected, the kinetic energy is very unstable due to the velocity being not as linear as predicted.

Afterwards, we changed the x-axis to position, creating a force vs. position graph. We took the integral of a part of the force vs. time plot in order to give us the kinetic energy of the cart. We also compared the actual kinetic energy based on the data calculations to the integral. The result looks like this:

We first took the integral of a small section of the force vs. position graph and found the value to be 0.09598 J, and the actual kinetic energy to be 0.132 J. We then took the integral of a larger area of the graph and found the value to be 0.2017 J, and the actual kinetic energy of the cart to be 0.269 J. The results were not as close as expected, and this may be due to the problems with leveling the track. It was expected that the kinetic energy of the cart is the same as the integral of the force vs. distance graph. Had we done better the leveling of the track, the results would have probably been even closer.

EXPT 2: Work Done by a Nonconstant Spring Force

Procedure: For this activity, we will be measuring the work done when we stretch a spring through a measured distance. Before we start, we first sketched an F vs. x graph constant force and an F vs. x graph with nonconstant force. Here are the sketches:

For both graphs, the area under the graph should be equal to the work done by the applied force because work by definition is force multiplied by time. Therefore, it makes sense for the work done on an object to be the area under the curve of a force vs. distance graph.

At the start of the experiment, we first calibrated the force sensor. Then, we set up the cart connected to a spring that is connected to a force sensor attached to a rod. We also set up the motion sensor at the back of the track. The set up looks something like this:

We set up Logger Pro by opening up a file with a preset setup of a force vs. position graph. We then zeroed the force probe and the motion detector with the cart next to the spring in an unstretched position. We also reversed the direction of the motion detector so that towards the detector is the positive direction. We also set up a formula within Logger Pro so that it would calculate the kinetic energy for us. We then began to graph the force vs position graph as the cart is moved slowly towards the motion detector until the spring is stretched to about 0.6 m. Here is a sketch of the graph:

We found the spring constant of the spring by looking at the slope of the graph, which turns out to be 2.206 +/- 0.01720. The spring constant is the slope of the graph because the equation of the spring force is F=kx, which resembles an equation of a line. Thus, the spring constant is the slope of the graph. We then used the integration routine in the software to find the work done in stretching the spring. Here is the result:

The work done in stretching the spring turns out to be 0.3130 J, according to the graph.

EXPT 3: Kinetic Energy And The Work-Kinetic Energy Principle

Procedure: For this part of the lab, we used the same set up as experiment 2. We zeroed out the force sensor and motion sensor at starting position, again, and then we stretched the cart with the spring attached to 0.6 m. Then we began graphing the graph once we let go of the cart and succumbed it to the spring force. Here is the resulting graph:

We used the integration routine in Logger Pro to find the area under the graph for three separate parts of the force vs. position plot. Here are the other two parts:

From this, we were able to calculate the change in kinetic energy and the work one on the cart. We summarized the data in a table:

Note that the sign of the work and change in kinetic energy is negative, and this is due to our "reversing" the direction when setting up the motion sensor.

Though our data does not show, due to the leveling of the track, the work done on the cart by the spring is supposed to be similar to its change in kinetic energy. And this goes according to the definition of the work-kinetic energy theorem, where work is equal to the change in kinetic energy. For this experiment, since the spring did some work on the cart, the spring changed its kinetic energy and increased its speed.

EXPT 4: Work-KE theorem

Procedure: For the last part of the lab, we watched a movie file that showed a professor using a machine to pull back on a large rubber band. While the machine pulled, the professor was also recording the force while the rubber band stretched a certain distance. Here is a sketch of the data:

Since the work done is the area under the graph, we found the work by splitting up the graph into its shapes and adding up all of the areas together. The shapes we used are a triangle, two rectangles, and a trapezoid. Here is the calculation:

According to the calculation, the work done pulling the rubber band is approximately 22.3 J. Afterwards, we calculated the final kinetic energy of the cart attached to the machine with the information we were given at the last few frames of the video. Here is the calculation for it:

According to this calculation, the final energy of the cart is approximately 23.8 J, slightly higher than what we calculated from the sketch.

Some sources of uncertainty for this part of the lab is the fact that we sketched the graph, rather than examined the raw data. Since the video depicted the graph as uneven and has many peaks and dives, the total work calculated might be slightly less than the actual work done. There was also a slight increase at the last part of the graph, but we sketched it as a straight horizontal line. This could have contributed to the slightly reduced number we calculated from the graph. Furthermore, there is also the method in which the professor recorded the data. She herself has to pull the paper across the surface, and this may have caused a bit of error to the data. There is also the device that stretched the rubber band. Perhaps the cart was in contact with friction and slightly reduced its kinetic energy. Overall, there were many sources of uncertainty, but the numbers are still relatively close, supporting the work-kinetic energy theorem.

Conclusion: The lab holistically showcases the work-kinetic energy in action . We understand that the change in kinetic energy can be used to calculate the work done on an object. Some areas of improvement could be replacing the track used, or leveling it better. Though the uncertainties exist on the experiments with the track, due to the leveling of the track, and our velocity is very skewed from its actual value (with an error of up to 70% at times), we learned through the last experiment that the change in kinetic energy is the same as the work done and how we can derive the spring constant from the graph of the spring force vs. position. We also learned that the area under the force vs. position graph is used to calculate the work done on an object, and by extension the change in kinetic energy.

5-April-2017: Lab 10: Activity -- Work and power

Eric Chong

Lab 10: Activity -- Work and Power

Lab Partners: Harvey Thai and Lynel Ornedo

Purpose: For this activity, we will be calculating our power output for lifting a known object and our power output for running up the stairs.

Introduction/Theory: Power is how fast one does work over time. Work depends on the direction of the force applied and how far it is applied. For this activity, we will be using the gravitational force as as the force and how high we go in order to measure how much work we do against gravity. We will be lifting a backpack with a known mass by pulling it up with a rope and timing it. We will also be running up stairs to see how fast we can do the work to reach the top height.

Procedure: We performed the backpack part of the lab first. For this part, we had a backpack with a known mass and we lifted it up to a certain height as fast as we could. The height that the bag reached is the same as the height of the top of the stairs in the second part of the lab, which turns out to be 4.498 m, and we will explain how we measured it in the next part. We used our phones to time how fast the lifting process was. Fairly simple. The time to lift the bag was 11.26 seconds.

For the next part, we ran up a set of stairs and timed how fast it took for us to reach the top. Again, we used our phones to time, and we estimated the angle that the stairs made with the ground to be around 30 degrees. We measured the height of one of the stairs, which turned out to be 17.3 cm, and multiplied it by 26, since there are 26 stairs. This means that the height at the top of the stairs is 4.498 m, as mentioned previously. We also used Harvey's mass, as he was the one who ran up the stairs, which turns out to be about 72.56 kg. We also recorded the time it takes to get up the stairs by walking instead of running. The time it takes for running is 5.46 seconds, and the time for walking is 13.29 seconds.

From here, we can calculate the power of both parts. We can use W=mgh, where m is mass in kg, g is gravitational acceleration, and h is height, for the work against gravity. After calculating the work, we can divide the work by time in order to get power. Here is the data with the calculations in spreadsheet form:

The first picture depicts the work done and the power of lifting the bag up to the specified height. The second picture shows the power of walking and running. As expected, there is less power in walking than in running, since the time for walking is larger.

Conclusion: a) Here are the calculations for the kinetic energy for walking:

Based on the kinetic energy calculated, compared to the work done when moving up against gravity, the value for kinetic energy is negligible, and thus will not produce a significant error in our calculations.

b) Here is the calculation for how many flight of stairs we have to run in order to equal the power output of a microwave oven

According to the calculation, the approximate number of stairs to run per second in order to match the power output of a microwave oven is 9 stairs.

c) Here is the calculation for the number of stairs to run in order to equal the amount of work it took to run the microwave:

According to the calculation, the number of stairs to run in order to equal the amount of work to run the microwave is approximately 3214 stairs.

d) 1.) Since the amount of work to heat water for a 10-minute shower is 12.5 MJ, the power is approximately 20833 Watts (calculated from converting 12.5 MJ to Joules, and the time to seconds, and dividing Joules by seconds).

2.) If I gathered a group of people to heat up my shower by riding bicycle-powered generators, I would need about 209 people (calculated from dividing 20833 by 100, since each person exhibits about 100 Watts of power).

3.) If I were going to provide all of the energy myself, I would need to ride for about 35 hours (calculated from dividing 12500000 J by 100 J/s, and then converting the result to hours).

Overall, this activity showcased the mechanics of power in relation to work and time. Some sources of uncertainty could stem from the reading of the timer due to reaction time, and also the measuring of the angle and the height of the stairs.