A 6-kg bowling ball rolling at 5 m/s strikes a stationary 4-kg bowling ball. If Ball #1 is moving forward at 2 m/s after the collision, what is the speed and direction of Ball #2? What is the impulse of the system? If the collision last for .5 seconds, how much force is exerted?

To find the speed and direction of Ball #2 after the collision, we can use the law of conservation of linear momentum. According to this law, the total momentum before the collision is equal to the total momentum after the collision.

Let's assume that the speed and direction of Ball #2 after the collision is v2.

Before the collision:
Momentum of Ball #1 = mass of Ball #1 * velocity of Ball #1 = 6 kg * 5 m/s = 30 kg·m/s

Momentum of Ball #2 = mass of Ball #2 * velocity of Ball #2 = 4 kg * 0 m/s (as it's stationary) = 0 kg·m/s

Total momentum before the collision = momentum of Ball #1 + momentum of Ball #2 = 30 kg·m/s + 0 kg·m/s = 30 kg·m/s

After the collision:
Momentum of Ball #1 = mass of Ball #1 * velocity of Ball #1 = 6 kg * 2 m/s = 12 kg·m/s

Momentum of Ball #2 = mass of Ball #2 * velocity of Ball #2 = 4 kg * v2 kg·m/s

Total momentum after the collision = momentum of Ball #1 + momentum of Ball #2 = 12 kg·m/s + 4v2 kg·m/s = 16v2 kg·m/s

According to the law of conservation of linear momentum, the total momentum before the collision is equal to the total momentum after the collision:

Total momentum before the collision = Total momentum after the collision
30 kg·m/s = 16v2 kg·m/s

Solving for v2:
v2 = 30 kg·m/s / 16 kg·m/s
v2 ≈ 1.875 m/s

So, the speed and direction of Ball #2 after the collision is approximately 1.875 m/s in the opposite direction of Ball #1.

To find the impulse of the system, we can use the formula:

Impulse = change in momentum

Before the collision, the momentum of the system is the sum of the momenta of the individual balls:

Initial momentum = momentum of Ball #1 + momentum of Ball #2
= 30 kg·m/s + 0 kg·m/s (as Ball #2 is stationary)
= 30 kg·m/s

After the collision, the momentum of the system is again the sum of the momenta of the individual balls:

Final momentum = momentum of Ball #1 + momentum of Ball #2
= 12 kg·m/s + 4v2 kg·m/s

Change in momentum = Final momentum - Initial momentum
= (12 kg·m/s + 4v2 kg·m/s) - 30 kg·m/s

Substituting the calculated value of v2:

Change in momentum = (12 kg·m/s + 4 * 1.875 kg·m/s) - 30 kg·m/s

Change in momentum ≈ -2.5 kg·m/s

Therefore, the impulse of the system is approximately -2.5 kg·m/s (the negative sign indicates a change in direction).

Finally, to calculate the force exerted during the collision, we can use the formula:

Impulse = force * time

Rearranging the formula:

Force = Impulse / time

Given that the impulse is -2.5 kg·m/s and the time is 0.5 seconds:

Force = -2.5 kg·m/s / 0.5 s

Force ≈ -5 N

Therefore, the force exerted during the collision is approximately -5 Newtons. The negative sign indicates a force in the opposite direction of motion.

To solve this problem, we will apply the principles of conservation of momentum and impulse.

1. **Speed and direction of Ball #2:**

Let's assume the speed and direction of Ball #2 after the collision is v2.

Using the principle of conservation of momentum, we know that the total momentum before the collision is equal to the total momentum after the collision.

Before collision:
Momentum of Ball #1 = mass of Ball #1 * initial velocity of Ball #1 = 6 kg * 5 m/s = 30 kg*m/s
Momentum of Ball #2 = mass of Ball #2 * initial velocity of Ball #2 = 4 kg * 0 m/s = 0 kg*m/s (as it is stationary)

After collision:
Momentum of Ball #1 = mass of Ball #1 * final velocity of Ball #1 = 6 kg * 2 m/s = 12 kg*m/s
Momentum of Ball #2 = mass of Ball #2 * final velocity of Ball #2 = 4 kg * v2 kg*m/s

Using the conservation of momentum, we have:
Total momentum before collision = Total momentum after collision
30 kg*m/s = 12 kg*m/s + 4 kg * v2 kg*m/s

Rearranging the equation, we get:
v2 = (30 kg*m/s - 12 kg*m/s) / 4 kg
v2 = 18 kg*m/s / 4 kg = 4.5 m/s

Therefore, the speed and direction of Ball #2 after the collision is 4.5 m/s, opposite to the direction of Ball #1 (since they move in opposite directions).

2. **Impulse of the system:**

Impulse can be calculated using the formula:
Impulse = Force * time

We know that impulse is equal to the change in momentum of an object. The impulse on an object can be calculated by multiplying the force acting on the object by the time interval over which the force acts.

Before the collision, both balls were at rest, so their initial momentum was zero. Therefore, the change in momentum is equal to the final momentum they acquire after the collision.

Change in momentum of Ball #1 = Mass of Ball #1 * (Final velocity of Ball #1 - Initial velocity of Ball #1)
= 6 kg * (2 m/s - 5 m/s) = -18 kg*m/s

Change in momentum of Ball #2 = Mass of Ball #2 * (Final velocity of Ball #2 - Initial velocity of Ball #2)
= 4 kg * (4.5 m/s - 0 m/s) = 18 kg*m/s

Since momentum is conserved, the change in momentum for Ball #2 is equal in magnitude but opposite in direction to the change in momentum for Ball #1.

Therefore, the impulse for the system of two balls is equal to the change in momentum of either ball:
Impulse = -18 kg*m/s (or 18 kg*m/s in magnitude)

3. **Force exerted during collision:**

Force can be calculated using the formula:
Force = Impulse / time

Given that the time of collision is 0.5 seconds, we can substitute the values:
Force = 18 kg*m/s / 0.5 s = 36 N

Therefore, the force exerted during the collision is 36 Newtons.