Two 0.4 kg soccer balls collide elastically in a head-on collision. The first ball starts at rest, and the second ball has a speed of 3.5 m/s. After the collision, the second ball is at rest.

A.) What is the final speed of the ball?
B.) What is the kinetic energy of the first ball before the collision?
C.) What is the Kinetic energy of the second ball after the collision?

Trick question

The collision was ELASTIC!!!
Therefore the energy after is the energy before and it is all kinetic.
Therefore:
If the moving ball stops, the formerly still ball goes on at the original speed.
A) 3.5 m/s
B) ZERO. The first ball was NOT moving before the collision.
The TRANSLATIONAL Ke of the moving ball is (1/2) (.4) (3.5)^2
This ignores the rotational Ke of a rolling ball. It is (1/2) I w^2 where I is the moment of inertia and w is the angular velocity, which we do not know.
C) The second ball was at REST after the collision so again ZERO.

A.) Well, after the collision, both balls end up at rest. So, the final speed of the ball is 0 m/s. It's like they hit each other and said, "Hey, let's take a break and have a nap!"

B.) Since the first ball is at rest, it doesn't have any kinetic energy before the collision. It's just chilling, enjoying the peace and quiet on the soccer field.

C.) After the collision, the second ball also comes to a stop. So, just like the first ball, it has no kinetic energy after the collision. They're both just sitting there, thinking about what they've done.

To solve this problem, we can use the principles of conservation of momentum and conservation of kinetic energy.

A.) The final speed of the ball can be found by applying the principle of conservation of momentum.

The momentum before the collision is given by:

Momentum1 + Momentum2 = 0

Since the first ball is at rest, its momentum is zero. The momentum of the second ball is calculated by multiplying its mass (0.4 kg) with its initial velocity (3.5 m/s):

Momentum2 = 0.4 kg * 3.5 m/s = 1.4 kg·m/s

After the collision, the second ball is at rest, so its momentum becomes zero. Therefore, the momentum after the collision is:

Momentum1 + Momentum2 = 0

Momentum1 = -1.4 kg·m/s

The final speed of the ball can be calculated by dividing the momentum by the mass of the ball:

Final Speed = Momentum1 / mass1 = -1.4 kg·m/s / 0.4 kg = -3.5 m/s

B.) The initial kinetic energy of the first ball can be found using the formula:

Kinetic Energy = 0.5 * mass * velocity^2

Since the first ball is at rest, its initial kinetic energy is zero.

C.) The kinetic energy of the second ball after the collision can also be found using the same formula:

Kinetic Energy = 0.5 * mass * velocity^2

Since the second ball is at rest after the collision, its kinetic energy is zero.

To solve these questions, we can use the principles of conservation of momentum and conservation of kinetic energy.

A.) To find the final speed of the ball (second ball after the collision), we can use the principle of conservation of momentum. In an elastic collision, the total momentum of the system before the collision must be equal to the total momentum after the collision.

Let's assume that the final speed of the second ball is v_f. Since the first ball is now at rest, its final velocity is 0 m/s.

Using the equation for conservation of momentum:
(mass1 * velocity1) + (mass2 * velocity2) = (mass1 * velocity1') + (mass2 * velocity2')

Where:
- mass1 is the mass of the first ball (0.4 kg)
- mass2 is the mass of the second ball (0.4 kg)
- velocity1 is the initial velocity of the first ball (0 m/s)
- velocity2 is the initial velocity of the second ball (3.5 m/s)
- velocity1' is the final velocity of the first ball (unknown)
- velocity2' is the final velocity of the second ball (0 m/s)

Plugging in the values:
(0.4 kg * 0 m/s) + (0.4 kg * 3.5 m/s) = (0.4 kg * velocity1') + (0.4 kg * 0 m/s)

0 + (0.4 kg * 3.5 m/s) = (0.4 kg * velocity1') + 0

1.4 kg m/s = 0.4 kg * velocity1'

Divide both sides by 0.4 kg to solve for velocity1':
velocity1' = (1.4 kg m/s) / 0.4 kg
velocity1' = 3.5 m/s

Therefore, the final speed of the ball (second ball after the collision) is 3.5 m/s.

B.) To find the kinetic energy of the first ball before the collision, we can use the formula for kinetic energy:

Kinetic energy = (1/2) * mass * velocity^2

Since the first ball is at rest, its initial velocity is 0 m/s.

Using the formula for kinetic energy:
Kinetic energy1 = (1/2) * mass1 * velocity1^2
Kinetic energy1 = (1/2) * 0.4 kg * (0 m/s)^2
Kinetic energy1 = 0 J

Therefore, the kinetic energy of the first ball before the collision is 0 J.

C.) To find the kinetic energy of the second ball after the collision, we again use the formula for kinetic energy:

Kinetic energy = (1/2) * mass * velocity^2

Since the second ball is at rest, its final velocity is 0 m/s.

Using the formula for kinetic energy:
Kinetic energy2 = (1/2) * mass2 * velocity2'^2
Kinetic energy2 = (1/2) * 0.4 kg * (0 m/s)^2
Kinetic energy2 = 0 J

Therefore, the kinetic energy of the second ball after the collision is 0 J.