Lab 9: Collision of Two Carts You will have to try to find the simulation on the screen-short in the attach files to complete the Lab.

The momentum and the kinetic energy of a moving particle are defined as

and ,

respectively, where is the mass of the particle and is its velocity.

The total momentum of a system of particles is the sum

,

while the total kinetic energy is defined as

.

In an isolated system, the total momentum is conserved during any dynamical evolution of the system, so that

.

In the simpler case of two colliding particle, if the total kinetic energy is also conserved, it is said that the two particles undergo an elastic collision. If there is any kinetic-energy loss, the collision is called inelastic. When the colliding particles stick together, forming one new object or moving as one after the collision, the collision is classified as perfectly inelastic.

The JavaScript provided with this document simulates the collision of two carts in one dimension, along a horizontal axis with a positive direction pointing to the right. It allows a user to change the initial velocities ( and ) and the masses ( and ) of the two colliding carts, and it shows the result as final velocities after the collision ( and ), with randomly built-in uncertainties. Get familiar with running the simulation by changing the different available setting and controls and observe their effects.

1. Choose “elastic” collision from the drop-down list at the top of the simulation.

2. Set the two masses to unequal values (ex. and ) and run the simulation five times for different initial velocities.

3. Record the values of and in the data table below.

4. For each step, record the initial velocities and you chose, and the final velocities and calculated by the JavaScript simulation.

5. Calculate the total initial and final momenta, and , and write their values in the table.

6. Calculate the total initial and final kinetic-energy values, and , and write them in the table.

7. Calculate the percentage differences in total momentum and kinetic energy and , respectively, and report them in the data table below.

1. Chose “perfectly inelastic” collision from the drop-down list at the top of the simulation.

2. Set the two masses to unequal values (ex. and ) and run the simulation five times for different initial velocities.

3. Record the values of and in the data table below.

4. For each step, record the initial velocities and you chose, and the final velocities and calculated by the JavaScript simulation.

5. Calculate the total initial and final momenta, and , and write their values in the table.

6. Calculate the total initial and final kinetic-energy values, and , and write them in the table.

7. Calculate the percentage difference in total momentum and the kinetic-energy loss and , respectively, and report them in the data table below.

1. Show mathematically that the kinetic energy of a moving particle with momentumcan be written as

.

1. Based on the data collected while using the JavaScript simulation, what can you say in general about the total momentum values before and after the collision of the two carts?

1. Based on the data collected while using the JavaScript simulation, what can you say in general about the total kinetic-energy values before and after the collision of the two carts?

1. A moving mass,, collides perfectly inelastically with a stationary mass, . Show that the total kinetic energy after the collision, , is equal to , where is the initial kinetic energy of this system.

1. A mass , with an initial velocity of 4 m/s, and a mass , initially at rest, undergo an elastic collision. Calculate their final velocities after the collision.

1. A mass , with an initial velocity of 4 m/s, and a mass , initially at rest, undergo a perfectly inelastic collision. Calculate the final velocity after the collision and the kinetic-energy loss.

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