A baseball (m = 148 g) approaches a bat horizontally at a speed of 43.8 m/s (98.0 mi/h) and is hit straight back at a speed of 48.4 m/s (108 mi/h). If the ball is in contact with the bat for a time of 1.08 ms, what is the average force exerted on the ball by the bat? Neglect the weight of the bat, since it is so much less than the force of the bat. Choose the direction of the incoming ball as the positive direction.

Why did the baseball bring a calculator to the game?

Because it wanted to calculate how much it would be "bat"-tered!

To find the average force exerted on the ball by the bat, we can use the impulse-momentum principle:

Impulse = change in momentum

Impulse = m * (vf - vi)

where m is the mass of the ball, vf is the final velocity of the ball, and vi is the initial velocity of the ball.

Given:
m = 148 g = 0.148 kg
vi = 43.8 m/s
vf = -48.4 m/s (since the ball changes its direction)

Impulse = 0.148 kg * (-48.4 m/s - 43.8 m/s)

Now, we need to find the change in velocity, which is equal to the final velocity minus the initial velocity.

Change in velocity = vf - vi

Change in velocity = -48.4 m/s - 43.8 m/s

Substituting values:

Impulse = 0.148 kg * (-48.4 m/s - 43.8 m/s)

Impulse = 0.148 kg * (-92.2 m/s)

Impulse = -13.6576 kg⋅m/s

Since impulse is equal to the average force multiplied by the time of contact, we can write:

Impulse = F * t

Solving for the average force:

F = Impulse / t

Given:
t = 1.08 ms = 1.08 × 10^(-3) s

F = (-13.6576 kg⋅m/s) / (1.08 × 10^(-3) s)

Calculating:

F ≈ -12,665.185 N (the negative sign indicates the opposite direction as the positive incoming ball)

Hence, the average force exerted on the ball by the bat is approximately 12,665.185 N in the opposite direction of the incoming ball.

To find the average force exerted on the ball by the bat, we can use Newton's second law of motion, which states that force (F) is equal to the rate of change of momentum (p).

The momentum of an object is defined as the product of its mass (m) and its velocity (v). In this case, we have the initial momentum of the ball (before it is hit) and the final momentum of the ball (after it is hit).

Initial momentum (p₁) = mass (m) * initial velocity (v₁)
Final momentum (p₂) = mass (m) * final velocity (v₂)

The change in momentum (Δp) is given by:
Δp = p₂ - p₁

Since force (F) is equal to Δp divided by the time (t) taken for the change in momentum to occur, we have:
F = Δp / t

Now let's calculate the change in momentum and the average force:

Step 1: Convert the mass of the ball from grams to kilograms:
m = 148 g = 148 / 1000 kg = 0.148 kg

Step 2: Convert the velocities from miles per hour to meters per second:
v₁ = 43.8 m/s
v₂ = 48.4 m/s

Step 3: Calculate the initial momentum and the final momentum:
p₁ = m * v₁
= 0.148 kg * 43.8 m/s

p₂ = m * v₂
= 0.148 kg * 48.4 m/s

Step 4: Calculate the change in momentum:
Δp = p₂ - p₁

Step 5: Calculate the average force:
F = Δp / t
= (p₂ - p₁) / t

Substituting the given value for the time, t = 1.08 ms = 1.08 × 10^(-3) s, we can now calculate the average force exerted on the ball by the bat.

a=(V-Vo)/t = (-48.4--43.8)/1.08=-85.37 m/s^2.

F = m*a = 0.148 * -85.37 = -12.6 N.