1. A 34.0-ounce baseball bat travels at 98.2 mph when it has a head-on collision with a 0.505-ounce ball traveling in the opposite direction at 84.5 mph. If the bat slows down to 95.7 mph after impact, what is the new velocity of the ball?

2. A 5235 lb car traveling north at 56.0 mph hits the rear-end of a 4845 lb car also going north at 46.0 mph. If the 4845 lb car is traveling at 58.8 mph after the collision, what is the velocity of the other car?

To find the new velocity of the ball after a head-on collision with a baseball bat, we can use the principle of conservation of momentum. By calculating the momentum of the ball and the bat before and after the collision, we can set up an equation to solve for the velocity of the ball.

Using the given information, the new velocity of the ball is approximately 67.72501 mph. For the second question, the velocity of the other car after the collision is approximately -41.062 mph.

To solve this problem, we can use the principle of conservation of momentum. The momentum before the collision is equal to the momentum after the collision.

We can calculate the momentum of the ball and the bat separately before the collision, and then set up an equation using the given information about the bat's velocity after the collision to find the velocity of the ball.

Let's start with the first question. The momentum before the collision is the sum of the momentum of the bat and the momentum of the ball. We can calculate their momentum using the formula p = mv, where p is momentum, m is mass, and v is velocity.

For the bat:

For the ball:

Now, let's calculate the momentum before the collision for both the bat and the ball:

The total momentum before the collision is:

Total momentum = Bat's momentum + Ball's momentum = 208.015 lbs·mph + (-2.66928125 lbs·mph) = 205.3457188 lbs·mph

The momentum after the collision is also the sum of the momentums of the bat and the ball. We can calculate their momentums using the same formula.

For the bat:

For the ball:

Now, let's calculate the momentum after the collision for both the bat and the ball:

Since the total momentum before the collision must be equal to the total momentum after the collision, we can set up the following equation:

Total momentum = Bat's momentum + Ball's momentum

205.3457188 lbs·mph = 203.2125 lbs·mph + Ball's momentum

Ball's momentum = 205.3457188 lbs·mph - 203.2125 lbs·mph

Ball's momentum = 2.1332188 lbs·mph

The velocity of the ball after the collision can be calculated using the formula v = p/m, where v is velocity, p is momentum, and m is mass.

For the ball:

Now, let's calculate the velocity of the ball after the collision:

Velocity = 67.72501 mph (approximately)

Therefore, the new velocity of the ball after the collision is 67.72501 mph.

For the second question, we can follow a similar process to find the velocity of the other car after the collision. The momentum before the collision is equal to the momentum after the collision.

For the first car:

For the second car:

Let's calculate the momentum before the collision for both cars:

Total momentum before the collision = First car's momentum + Second car's momentum = 293160 lbs·mph + (-222870 lbs·mph) = 70290 lbs·mph

For the first car:

For the second car:

Now, let's calculate the momentum after the collision for both cars:

Setting up the equation using the total momentum before the collision:

Total momentum = First car's momentum + Second car's momentum

70290 lbs·mph = ? + 285132 lbs·mph

First car's momentum = 70290 lbs·mph - 285132 lbs·mph

First car's momentum = -214842 lbs·mph

For the first car:

Now, let's calculate the velocity of the first car after the collision:

Velocity = -41.062 mph (approximately)

Therefore, the velocity of the first car after the collision is -41.062 mph.

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