A catcher stops a 48.5 m/s pitch in his glove, bringing it to rest in 0.15 m. If the force exerted by the catcher is 803 N, what is the mass of the ball?
F=ma=mv²/2s
m=2Fs/v²
.14
To find the mass of the ball, we can use Newton's second law of motion, which states that the force applied to an object is equal to the product of its mass and acceleration. In this case, the acceleration is due to the deceleration of the ball when it comes to rest in the catcher's glove.
First, let's calculate the acceleration of the ball using the formula of acceleration:
acceleration = (final velocity - initial velocity) / time
Here, the initial velocity (v_0) is 48.5 m/s, the final velocity (v) is 0 m/s (as the ball comes to rest), and the time (t) is not given. However, we can assume that the pitch stops within a very short time interval (we neglect the reaction time of the catcher). Therefore, we can consider the time to be very small or approximately equal to zero.
acceleration = (0 m/s - 48.5 m/s) / 0s
acceleration = -48.5 m/s / 0s
Here, we must be careful because the denominator is zero, which is undefined. However, we can consider the limit of the acceleration as the time approaches zero.
In this case, the acceleration becomes infinite.
As the acceleration is very high and time is very small, the force exerted can bring the ball to rest within a short distance. Since the catcher applies the force over a distance of 0.15 m, the work done by the catcher (force times distance) is equal to the change in kinetic energy of the ball.
work done = change in kinetic energy
The initial kinetic energy of the ball (K_0) is given by:
K_0 = (1/2) * mass * initial velocity^2
And the final kinetic energy (K) is zero as the ball comes to rest.
Therefore, we can write the equation:
work done = K - K_0
803 N * 0.15 m = 0 - (1/2) * mass * (48.5 m/s)^2
Simplifying the equation:
803 N * 0.15 m = -(1/2) * mass * (48.5 m/s)^2
Now, we can solve for the mass of the ball:
mass = [803 N * 0.15 m] / [-(1/2) * (48.5 m/s)^2]
mass ≈ 0.0101 kg
Therefore, the mass of the ball is approximately 0.0101 kg.