Question

What average force is necessary to stop a bullet of mass 20g and speed 250 m/s as it penetrates wood to distance of 12 cm:
(A) $ 3.4 \times {10^3}newton $
(B) $ 5.2 \times {10^3}newton $
(C) $ 3.4 \times {10^5}newtons $
(D) $ 5.2 \times {10^5}newton $

Answer 1

Hint: The average force would be the force required to decelerate the bullet from 250 m/s to zero. To find the force we need to calculate the acceleration (actually deceleration or retardation). And then using its value and the value of the mass of the bullet we can calculate the force.

Formula used: In this solution we will be using the following formula;
 $ F = ma $ where $ F $ is the force acting on an object, $ m $ is the mass of that body, and $ a $ is the acceleration of the body.
 $ {v^2} = {u^2} \pm 2as $ where $ v $ is the final velocity, $ u $ is the initial velocity and $ s $ is the distance travelled for the body to accelerate from initial to final velocity.


Complete step by step answer
A bullet is said to be moving at 250 m/s. the force required to stop the bullet if it moves through a distance of 12 m before coming to rest. Hence, using one of the equations of motion, the deceleration the stone must acquire must be determined.
One of the equations can be given as,
 $ {v^2} = {u^2} \pm 2as $, hence since the object is decelerating, then
 $ {v^2} = {u^2} - 2as $ where $ v $ is the final velocity, $ u $ is the initial velocity and $ s $ is the distance travelled for the body to accelerate from initial to final velocity and $ a $ is the deceleration.
Inserting all known values, we have that
 $ 0 = {250^2} - 2a\left( {0.12} \right) $
By computation, we have
 $ 0 = 62500 - 0.24a $
 $ \Rightarrow 0.24a = 62500 $
By dividing by $ 0.24 $ we have that
 $ a = 26417m/{s^2} $
Then we can calculate for the force by using the Newton’s second law equation as in
 $ F = ma $ where $ F $ is the force acting on an object, $ m $ is the mass of that body, and $ a $ is the acceleration of the body.
Hence,
 $ F = 0.02\left( {26417} \right) $
By multiplying
 $ F = 5208N = 5.2 \times {10^3}N $

Hence, the correct answer is B.

Note
For clarity, the acceleration is made negative because it occurs in the opposite direction as the direction of the velocity, hence if the direction of the velocity is considered positive, the direction of acceleration must be considered negative.
In application, similar but more complex kinds of calculations are done when bulletproof materials are to be made. The force required to stop the bullet must be investigated, then checked against the strength (like shear modulus and all) of the material.