Estimate the average force that a baseball pitcher's hand exerts on a 0.145-kg baseball as he throws a 39-m/s pitch. Assume the pitcher accelerates the ball through a distance of 3.0 m, from behind his body to where the ball is released and ignore the air resistance.
I just want the answer.
To estimate the average force that a baseball pitcher's hand exerts on a baseball, we can use the concept of impulse.
Impulse is defined as the product of the force applied to an object and the time interval over which it was applied. Mathematically, impulse can be expressed as:
Impulse = Force * Time
In this case, the time interval is not given. However, we can find it using the given information.
The pitcher accelerates the ball from rest to a speed of 39 m/s over a distance of 3.0 m. To find the time interval, we can use the formula for acceleration:
Vf^2 = Vi^2 + 2aΔd
Here, Vf is the final velocity, Vi is the initial velocity (which is 0 as the ball starts from rest), a is the acceleration, and Δd is the distance.
Plugging in the values, we get:
(39 m/s)^2 = 0 + 2a * 3.0 m
a ≈ (39 m/s)^2 / (2 * 3.0 m)
Simplifying this expression, we find that the acceleration is approximately 254.25 m/s^2.
Now, we can use the formula for average force:
Average Force = Impulse / Time
Since the impulse is equal to the change in momentum and the time interval is the time it takes to accelerate the ball, which is the same as the time taken to traverse the distance of 3.0 m, we can rewrite the formula as:
Average Force = (Δp) / t
Since the mass of the baseball is given as 0.145 kg and the final velocity of the ball is 39 m/s, we can calculate the change in momentum (Δp) as:
Δp = m * (Vf - Vi)
Δp = 0.145 kg * (39 m/s - 0 m/s)
Δp ≈ 5.655 kg·m/s
Finally, we divide the change in momentum by the time interval to find the average force:
Average Force = 5.655 kg·m/s / t
Now, we need to find the time interval (t). We can use the equation for average speed:
Average Speed = Total Distance / Total Time
The distance traveled by the ball is 3.0 m, and the average speed is given as 39 m/s. We can rearrange the equation to find the total time (t):
t = Total Distance / Average Speed
t = 3.0 m / 39 m/s
t ≈ 0.07692 s
Now, we can substitute this time value back into the equation for average force:
Average Force = 5.655 kg·m/s / 0.07692 s
Calculating this expression, we find that the average force exerted by the pitcher's hand on the baseball is approximately 73.34 Newtons.