Question

A30 gbullet moving a horizontal velocity of500 m/scomes to a stop 12 cmwithin a solid wall. (a) What is the change in the bullet’s mechanical energy? (b) What is the magnitude of the average force from the wall stopping it?

Short answer

1. Change in the bullet’s mechanical energy will be $-3.8\mathrm{x}{10}^3\mathrm{J}$.
2. Magnitude of the average force from the wall to stop the bullet will be $3.1\times {10}^4\mathrm{N}$.

Step-by-step solution

Given data:

Mass of a bullet, $m=30g=0.030kg$

Distance, $d=12cm=0.12m$

A horizontal initial velocity, $v_i=500m/s$

Final velocity, $v_f=0m/s$

To understand the concept:

As the bullet is moving in the horizontal direction, there will be no change in the potential energy of the bullet. Hence, you can neglect it. Using the change in kinetic energy of the bullet, you can find the change in the bullet’s mechanical energy.

As known that work done by the force is equal to the change in mechanical energy. Using this, you equate this to the formula for work to find the required force.

Formulae:

The change in kinetic energy is,

$∆KE=K{E}_{final}-K{E}_{initial}$

Here, $K{E}_{initial}andK{E}_{final}$and are the initial and final kinetic energy respectively.

The kinetic energy is defined by,

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

Write the work done equations as below.

$W=F∆\mathrm{d}$ ….. (1)

And

$W=\left|∆E_{mech}\right|$ ….. (2)

Here, W is the work done, F is the force, $∆d$ is the change in distance, and $∆E_{mech}$ is the change in mechanical energy.

(a) The change in the bullet’s mechanical energy:

Calculate the initial kinetic energy as below.

$$\begin{gathered} K{E}_{initial}=\frac{1}{2}m{v}_i^2 \\ =\frac{1}{2}\times 0.03\mathrm{kg}\times {\left(500\mathrm{m}/\mathrm{s}\right)}^2 \\ =3750\mathrm{J} \\ =3.8\times {10}^3\mathrm{J} \end{gathered}$$

Determine the final kinetic energy as below.

$$\begin{gathered} K{E}_{final}=\frac{1}{2}m{v}_f^2 \\ =\frac{1}{2}\times 0.03kg\times {(0m/s)}^2=0J \end{gathered}$$

Therefore, the change in kinetic energy is,

$$\begin{gathered} ∆KE=K{E}_{final}-K{E}_{initial} \\ =0-3.8\times {10}^3\mathrm{J} \\ =-3.8\times {10}^3\mathrm{J} \end{gathered}$$

As the ball travels in the horizontal direction, there is no change in potential difference.

$$\begin{gathered} ∆E_{mech}=∆KE \\ =-3.8\times {10}^{3}J \end{gathered}$$

Hence, the change in the bullet’s mechanical energy will be $-3.8\times {10}^3\mathrm{J}$.

(b) Calculate the magnitude of the average force from the wall stopping it

From equation (1) and (2), you can write the following equation.

$$\begin{gathered} F∆d=|∆E_{mech}| \\ F=\frac{\left|∆E_{mech}\right|}{d} \\ =\frac{3.8\times {10}^{8}J}{0.12m} \\ =3.1\times {10}^{4}N \end{gathered}$$

Hence, the magnitude of the average force from the wall to stop the bullet will be $3.1\times {10}^{4}N$.