A major-league catcher gloves a 92 mi/h pitch and brings it to rest in 0.15 m.
1-If the force exerted by the catcher is 803 N, what is the mass of the ball?
0.14
To solve this problem, we can use Newton's second law of motion, which states that the force exerted on an object is equal to the mass of the object multiplied by its acceleration:
F = m * a
In this case, the force exerted by the catcher on the ball is 803 N. Since the ball comes to rest, its final velocity is 0 m/s. We can calculate the initial velocity of the ball using the formula:
v^2 = u^2 + 2as
Where v is the final velocity (0 m/s), u is the initial velocity, a is the acceleration, and s is the distance traveled.
Given that the speed of the pitch is 92 mi/h, we need to convert it to m/s:
92 mi/h * (1609 m / 1 mi) * (1 h / 3600 s) = 41.2 m/s
Now we can substitute the values into the formula to find the initial velocity:
0^2 = u^2 + 2 * a * s
0 = u^2 + 2 * -a * 0.15
0 = u^2 - 0.3a
u^2 = 0.3a
u = sqrt(0.3a)
Since the acceleration is negative (opposite to the initial velocity), we should consider it as -9.8 m/s^2 (acceleration due to gravity).
Plugging in the numbers, we can solve for the initial velocity:
sqrt(0.3 * (-9.8 m/s^2)) = -1.642 m/s (rounded to three decimal places)
Now that we know the initial and final velocities, we can calculate the acceleration using the formula:
a = (v - u) / t
Where the final velocity v is 0 m/s, the initial velocity u is -1.642 m/s, and the time t is the time taken to come to rest, which is not given. However, we don't need to calculate it since it cancels out when solving for the mass.
Now, we can rearrange the formula F = m * a to solve for the mass:
m = F / a
m = 803 N / -9.8 m/s^2
Calculating the mass:
m ≈ -82 kg (rounded to two decimal places)
Note: The negative sign indicates that the acceleration and mass have opposite directions, which is expected since the ball is decelerating (slowing down) in the opposite direction to the initial motion.