How much force does a baseball pitcher have to exert on a 250g baseball to make it accelerate to 50m/s the instant that it leaves his hand ?
To calculate the force required, we can use Newton's second law of motion, which states:
Force (F) = mass (m) x acceleration (a)
Given:
Mass of the baseball (m) = 250 g = 0.25 kg
Velocity of the baseball (v) = 50 m/s
Time taken for the baseball to reach this velocity (t) = 0 seconds (instantaneous)
Acceleration (a) can be calculated using the formula:
v = u + at
Given:
Initial velocity (u) = 0 (as the baseball starts from rest)
Final velocity (v) = 50 m/s
Time (t) = 0 s
Using the above information in the equation, we can find the acceleration (a):
50 = 0 + a * 0
50 = 0
a = 50/0
Since the equation becomes undefined (division by zero), we need to find another approach.
An alternative way to calculate the force is by using the impulse-momentum equation, which states:
Impulse (J) = Force (F) x time (Δt)
Impulse is the change in momentum. Since the initial momentum is zero (as the baseball starts from rest), the impulse is equal to the final momentum.
Impulse (J) = momentum (p) = m * v
Given:
Mass of the baseball (m) = 0.25 kg
Velocity of the baseball (v) = 50 m/s
Impulse (J) = m * v
Impulse (J) = 0.25 kg * 50 m/s
Impulse (J) = 12.5 kg·m/s
Now, we need to find the time (Δt) it takes for the baseball to accelerate to this velocity.
Given:
Initial velocity (u) = 0 (as the baseball starts from rest)
Final velocity (v) = 50 m/s
The time (Δt) can be calculated using the formula:
v = u + at
50 = 0 + a * t
50 = a * t
We know the acceleration (a) is non-zero but don't have its value. However, we can assume a value of acceleration since the question doesn't provide it. Let's assume a = 10 m/s² for demonstration purposes.
50 = 10 * t
t = 50 / 10
t = 5 seconds
Now that we have the impulse (J) and the time (Δt), we can calculate the force (F):
Impulse (J) = Force (F) x time (Δt)
12.5 kg·m/s = F * 5 s
12.5 kg·m/s = 5F
Solving for F:
F = 12.5 kg·m/s / 5 s
F = 2.5 Newtons
Therefore, the pitcher needs to exert a force of 2.5 Newtons on the 250g baseball to make it accelerate to 50 m/s the instant it leaves his hand, assuming an acceleration of 10 m/s².