11.1.7.2: Explorations

Exploration 1: Push a Cart Around

The cart in the animation interacts with the two-handed image if the image is near the left-hand or right-hand end of the cart (position is given in meters and time is given in seconds). Move the image from side to side and observe the resulting graph. The arrow below the cart shows the direction and strength of the force. Restart the animation if the cart goes off screen. Restart.

Define the system to be just the cart and answer the following questions assuming that you are moving the cart around with the "handy" image.

1. Is the energy of the system constant? If not, where is it coming from?
2. Does the energy always decrease if the image is to the right of the cart? Does it increase if the image is to the left of the cart?

Exploration 2: Choice of Zero for Potential Energy

The animation depicts a ball being dropped from \(y = 15\text{ m}\) onto the ground \(15\) meters below at \(y = 0\text{ m}\) (position is given in meters and time is given in seconds). For this animation we will assume that the ball undergoes a very hard collision with the ground, which also conserves energy. Also shown are two pairs of bar graphs representing the different types of energy associated with the ball: the kinetic energy (orange) and the gravitational potential energy (blue). The bar graphs on the left show the kinetic energy and the potential energy as measured from \(y_{\text{ref}} = 0\text{ m}\). The bar graphs on the right show the kinetic energy and the potential energy with a varying zero potential energy point. You can vary the zero point from \(-15\text{ m} < y_{\text{ref}} < 15\text{ m}\) by changing the value in the text box and clicking the "set value and play" button. Restart.

Change the zero point for the potential energy from zero to a variety of positive values and a variety of negative values. Answer the following questions about the animation.

1. For zero points that are less than zero, does the gravitational potential energy shift up or down?
2. Is all of this energy accessible to the ball? In other words, can it all be converted to kinetic energy?
3. For zero points that are greater than zero, does the gravitational potential energy shift up or down?
4. For \(y_{\text{ref}} = -15\text{ m}\), how much potential energy does the ball start out with? How much does it have when it hits the ground? What is the change in potential energy?
5. For \(y_{\text{ref}} = 15\text{ m}\), how much potential energy does the ball start out with? How much does it have when it hits the ground? What is the change in potential energy?
6. How do your answers for (d) and (e) compare? Why?

Exploration 3: Elastic Collision

The initial velocities of the two carts in the above animation can be changed by entering new values into the text fields (position is given in meters, time is given in seconds, and energy on the bar graph is given in joules). As the carts approach one another, they begin to repel due to the magnets carried by each of them, thereby changing their velocities. The two color-coded bar graphs on the right show the instantaneous kinetic energy of the carts. Restart.

When you get a good-looking graph, right-click on it to clone the graph and resize it for a better view.

1. Run the animation using \(2\text{ m/s}\) and \(-2\text{ m/s}\) for the velocities of the left and right carts, respectively. What is the change in kinetic energy of the left cart? The right cart? What is the total change in energy?
2. Simulate collisions using other values of equal but opposite velocities. How does this effect each cart's change in kinetic energy? The change in the total energy?
3. Stop the animation just as the collision is about to take place and step forward in time so that the animation is paused during the collision process. What happens to the total energy during the collision process?
4. Does the last result imply that the two-cart system is not isolated?
5. Run the animation using \(1\text{ m/s}\) and \(-2\text{ m/s}\) for the velocities of the left and right carts, respectively. What is the change in total kinetic energy produced by the collision?

Exploration 4: A Ball Hits a Mass Attached to a Spring

Whenever objects interact, energy is likely to be converted from one form to another and/or dissipated (position is given in meters and time is given in seconds). Consider two models of a ball hitting a \(0.4\text{-kg}\) rectangle attached to a massless spring. After the collision the masses stick together and oscillate. Animation 1 represents an ideal spring and frictionless conditions, while Animation 2 represents a more realistic spring, and friction takes its inevitable toll on the system (only the kinetic energy of the ball is shown in the graph). Consider a system made up of the mass, the rectangle, and the massless spring as you answer the following questions. The potential energy of the spring is zero when the spring is uncompressed and, since the spring is massless, it has no kinetic energy. Restart.

1. What is the mass of the black ball?
2. What is the initial energy of the system?

Answer the following questions for each animation.

3. What is the energy of the system immediately after the collision?
4. Draw energy diagrams for the three objects that make up the system at the following times: \(t = 0\text{ s}\), \(t = 1.90\text{ s}\), \(t = 4.10\text{ s}\), \(t = 6.30\text{ s}\), and \(t = 8.55\text{ s}\).
5. For Animation 2 only: Approximately how long does it take for \(80\%\) of the initial energy to be dissipated?

Exploration 5: Drag the Ball to Determine \(PE(x)\)

Potential energy is energy associated with the configuration of an object or a system (position is given in meters and time is given in seconds). Since potential energy can be converted to kinetic energy, an operational way to determine potential energy is to let the system evolve from an unknown configuration to a known configuration and measure the kinetic energy. You can use this technique to measure and plot potential energy functions, \(PE(x)\). Restart

Plot the potential energy as a function of position for both animations. Note that these interactions may or may not be physical interactions.

Procedure: Reset will initialize the system to a known potential energy. This initial configuration has been marked with a small red dot. Assume this configuration has zero potential energy, \(PE_{0} = 0\), and the object has a mass of \(1\text{ kg}\). Use the mouse to move the object to a new position and release it. The object will have zero initial velocity when it is released. If the object returns to the original position you can record the velocity and calculate the kinetic energy. This kinetic energy must have come from the potential energy at the new position if the interaction is conservative.

Note

Animation will stop after \(100\text{ s}\).

Exploration 6: Different Interactions

The animations show a red ball that you can drag with the mouse (position is given in meters, time is given in seconds, and energy on the bar graph is given in joules). The bar graph shows the negative of the force on the ball integrated over the displacement from the origin. This is the negative of the work done on the ball to get it to this position. This integral, when the force is conservative, is also the potential energy associated with the ball when it is at this position. Also shown is a table with a calculation of position and the negative of the work. Restart.

1. Briefly describe the force in each animation.
2. Which of the forces is conservative? Why?
3. For the conservative forces, draw the potential energy function.

Exploration 7: Exploring Potential Energy Functions

Select a possible potential energy function. Drag the crosshair cursor with the mouse. The bar graph on the right displays the work done along the path by the force that corresponds to the given potential energy function. For your reference, there are circles every \(10\text{ m}\) that form a coordinate grid (position is given in meters and the result of the integral given on the bar graph is in joules). Restart.

1. Describe each potential energy function in words.
2. How does the work relate to the change in potential energy along a certain path?
3. What happens when you drag the cursor through a closed path (a path that begins and ends at the same point)?
4. What is the force that is responsible for each potential energy function? Write the force in the \(x\) and \(y\) direction as a function of \(x\) and \(y\), \(F_{x}(x, y)\) and \(F_{y}(x, y)\).

When you are finished with this Exploration, feel free to enter your own potential energy function.

Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.