Question

A baseball (m=145g) traveling 34 m/s moves a fielder's glove backward 26 cm when the ball is caught. What was the average force exerted by the ball on the glove?

Answer 1

Concepts and reason

The concepts that are to be used to solve the given problem are work-energy theorem, work, and kinetic energy.

First, express the work done as force multiplied by the distance along the force. Then, express the initial and final kinetic energies of the object. Next, use work energy theorem to equate the work done to kinetic energy change. Finally, solve the resulting equation for force and substitute the values in to the derived expression for force to determine the amount of force exerted on the glove by the ball.

Fundamentals

Express the kinetic energy K of the object of mass m moving with speed v as follows:

$K = \frac{1}{2}m{v^2}$

The work done W by the force F is,

$W = Fd$

Here, $d$ is the distance moved along the force.

The initial kinetic energy ${K_i}$ of the ball is,

${K_i} = \frac{1}{2}m{v_i}^2$

Here, ${v_i}$ is the initial speed and m is the mass of the ball.

Express the final kinetic energy ${K_f}$ of the ball as follows:

${K_f} = \frac{1}{2}m{v_f}^2$

Here, ${v_f}$ is the final speed of the ball.

Express the work done by the force as follows:

$W = - Fd$

From work energy theorem:

$W = {K_f} - {K_i}$

Replace $W$ with $- Fd$ , ${K_i}$ with $\frac{1}{2}m{v_i}^2$ , and ${K_f}$ with $\frac{1}{2}m{v_f}^2$ in the above equation and solve for F.

$\begin{array}{c}\\ - Fd = \frac{1}{2}m{v_f}^2 - \frac{1}{2}m{v_i}^2\\\\F = \frac{{m\left( {{v_i}^2 - {v_f}^2} \right)}}{{2d}}\\\end{array}$

Substitute 145 g for m, 26 cm for d, 34 m/s for ${v_i}$ , and 0 m/s for ${v_f}$ in the equation $F = \frac{{m\left( {{v_i}^2 - {v_f}^2} \right)}}{{2d}}$ .

$\begin{array}{c}\\F = \frac{{\left( {145{\rm{ g}}\left( {\frac{{1{\rm{ kg}}}}{{1000{\rm{ g}}}}} \right)} \right)\left( {{{\left( {34{\rm{ m/s}}} \right)}^2} - {{\left( 0 \right)}^2}} \right)}}{{2\left( {26{\rm{ cm}}\left( {\frac{{1{\rm{ m}}}}{{100{\rm{ cm}}}}} \right)} \right)}}\\\\ = 322.3{\rm{ N}}\\\end{array}$

Ans:

The force exerted by the ball is $322.3{\rm{ N}}$ .