# 2. Problems

## 2.1. 1-D collision with ground I

A spherical object with mass m with radius r is released from a height ho above the ground with an initial vertical velocity of voy. When the object strikes the ground, it has an elastic collision.

• Choose values of ho, voy, and r and draw an approximate sketch of what the curves for the height and velocity of the object as a function of time should look like. Label any key points in terms of the parameters that you chose.
• Write down the equations for the exact solutions for y(t) and vy(t) before the object strikes the ground and then write a program to plot these equations.
• Write down equations for dy / dt and dvy / dt.
• The Forward Euler method requires a parameter Δt to be used. Instead of using trial-and-error, use the physics of the problem to come up with a Δt that is reasonable to use.
• Use the Forward Euler method for your equations for dy / dt and dvy / dt to estimate y(t) and vy(t) from the time of release until the object strikes the ground.
• Plot the exact and Forward Euler solutions for y(t) and vy(t) from the time the object is released to the time the object strikes the ground. One plot should show the exact and Forward Euler y(t) and the other plot should show the exact and Forward Euler vy(t).

Your code will be tested using the value of Δt that you chose and then several other values. Make sure that the solution makes sense at the reflection point. A common error is that the exact solution is not correct for arbitrary values of Δt (exact solution should not depend on this parameter - it is only relevant for Euler method solution!).

## 2.2. 1-D collision with ground II

• Write down the exact equations for y(t) and vy(t) after the object strikes the ground for the first time and before it strikes the ground for the second time.
• Use the Forward Euler method to estimate y(t) and vy(t) from the time of release until it strikes the ground for the second time.
• Plot the exact and Forward Euler solutions for y(t) and vy(t) from the time the object is released to the time that it hits the ground for the second time. One plot should show the exact and Forward Euler y(t) and the other plot should show the exact and Forward Euler vy(t).

## 2.3. 1-D collision with wall

This problem is similar to the previous problem except that the object does not experience the force of gravity.

An hockey puck of mass m, radius r, and velocity vox is moving along the x-axis towards a wall with which it has an elastic collision. Initially it is a distance L from the wall. There is no friction between the puck and the ice.

• Choose values of L, r, m, and vox and draw an approximate sketch of what the curves for the position and velocity of the puck as a function of time should look like from its time of release until it returns to its starting point. Label any key points.
• Write down the exact equations for y(t) and vy(t) before the puck strikes the wall.
• Write down equations for dx / dt and dvx / dt of the puck.
• Use the Forward Euler method and the equations you wrote down for dx / dt and dvx / dt to estimate x(t) and vx(t) before the puck strikes the wall.
• Plot the exact and Forward Euler solutions for x(t) and vx(t) before the puck strikes the wall.
• Write down the exact equations for x(t) and vx(t) after the puck strikes the wall.
• Use the Forward Euler method to estimate x(t) and vx(t) after the puck strikes the wall.
• Plot the exact and Forward Euler solutions for x(t) and vx(t) after the puck strikes the wall.

## 2.4. 1-D Collision with walls

You are in a room with a ceiling height of 2 meters. You throw a rubber ball with a radius of 0.1 meters straight downward with a velocity of 10 m/s. The ball is released from your hand at a height of 0.5 meters. The ball has elastic collisions with the floor and ceiling.

Use the Forward Euler method to compute the height and velocity of the ball as a function of time. The last time plotted should be at about the time the ball hits the floor for the third time.

When your program is executed, two figures should appear. The first should show y(t) and the second should show v(t).

## 2.5. 1-D collision between two masses

Repeat problem 2. but replace the wall with a puck of mass M and radius R that is initially at rest. Select your own values for the parameters.

## 2.6. 2-D collision with ground

Repeat problem 1. except give the object an initial non-zero horizontal velocity of vox

## 2.7. 2-D collision with wall

An hockey puck of mass m, radius r, and velocity vo is moving towards a wall at an angle of 45o. Initially it is a distance L from the wall. There is no friction between the puck and the ice and the collision is elastic. Compute and plot the exact and Forward Euler solutions for x, y, vx, and vy of the the puck from the initial time until the puck is a distance L from the wall.

## 2.8. 2-D collision between two masses

An hockey puck of mass m, radius r, and velocity vo is moving towards the origin at an angle of 45o. A second stationary puck of mass M and radius R is at the origin. Initially the moving puck is a distance L from the origin. There is no friction between the puck and the ice and the collision is elastic. Compute and plot the exact and Forward Euler solutions for x, y, vx, and vy of both pucks from the initial time until the pucks are separated by a distance L.