Question

A bullet fired into a target loses half of its velocity after penetrating 15 cm. How much further will it penetrate before coming to rest?

Answer 1

Hint: In this question we will use the equation of motion, which gives us relation between final velocity, initial velocity, acceleration and distance. Now, by substituting and solving the equation, we will get the required result. Further, we will discuss the basics of equations of motion.
Formula used:
${v^2} = {u^2} + 2as$

Complete step by step solution:
As we know, we have three equations of motion. So, by applying the equation of motion for the initial velocity of bullet, we get:
${v^2} = {u^2} + 2as$
Now, solving the above equation for acceleration a, we get:
$a = - \dfrac{{{u^2}}}{{40}}$
As we know, we have three equations of motion. So, by applying the equation of motion for bullet after penetrating 15cm, we get:
${v^2} = {u^2} + 2as$
Since, u=0, where the bullet is at rest
$v = \dfrac{u}{2}$
Now, applying these values in the equation of motion, we get:
${v^2} = {u^2} + 2as$
$\eqalign{& \Rightarrow \dfrac{{{u^2}}}{{20}}s = \dfrac{{{u^2}}}{4} \cr
  & \therefore s = 5cm \cr} $
Therefore, we get the required result that gives us the distance travelled by the bullet further before coming to rest.

Additional information:
As we know that the equations of motion are equations which describe the behavior of a physical system in terms of its motion as a function of time. Further we can say that these equations of motion describe the behavior of a physical system as a set of mathematical functions in terms of dynamic variables. Here, dynamic variables are said to be normally spatial coordinates and time is used, but others are also possible, like momentum components and time.
Now, if we go in history, these equations of motion were discovered by Galileo Galilee but he could not manage to prove it practically that his equations were right or not. Later, Sir Isaac Newton proved these three equations of motion practically and also graphically. So, that is the reason now they are often called Newton’s three equations of motion. These equations tell us about the acceleration, displacement, time, final velocity of an object, initial velocity of an object.

Note:
Here we should remember that the three different equations of motion are used in finding different physical properties of a particle under motion. We should also observe that these equations are only applicable to the classical system not in the quantum system.