Question

What is the recoil velocity of the gun of mass 8 kg when a bullet of mass 10 g is fired from it with a velocity of 400 m/s?
A. 5 m/s
B. 2 m/s
C. 50 m/s
D. 0.5 m/s

Answer 1

Hint: To solve this question, we use the basic theory of Conservation of linear momentum. First, we calculate the total initial momentum of the rifle and bullet system and then after calculate total momentum of the rifle and bullet system after firing and by using Conservation of linear momentum, we equate these two as discussed below.

Complete Step-by-Step solution:
Given in question,
Mass of the rifle, ${{\text{m}}_{\text{1}}}$= 8 kg
Mass of the bullet, ${{\text{m}}_2}$= 10g= 0.01 kg
Recoil velocity of the rifle= ${{\text{v}}_{\text{1}}}$
Bullet is fired with an initial velocity, ${{\text{v}}_2}$= 400 m/s
Initially, the rifle is at rest.
Thus, its initial velocity, v= 0
Initial momentum of the system (rifle and bullet) = (${{\text{m}}_{\text{1}}}$+${{\text{m}}_2}$) v= 0
Total momentum of the system (rifle and bullet) after firing:
${{\text{m}}_{\text{1}}}{{\text{v}}_{\text{1}}}{\text{ + }}{{\text{m}}_{\text{2}}}{{\text{v}}_{\text{2}}}$= ${\text{8}}{{\text{v}}_{\text{g}}}{\text{ + (0}}{\text{.01)(400)}}$
Now, apply Conservation of linear momentum:
Initial momentum (rifle and bullet) = Total momentum (rifle and bullet) after firing:
That means: ${{\text{P}}_{\text{i}}}{\text{ = }}{{\text{P}}_{\text{f}}}$
$ \Rightarrow $ 0 = ${\text{8}}{{\text{v}}_{\text{g}}}{\text{ + (0}}{\text{.01)(400)}}$
$ \Rightarrow $ ${{\text{v}}_{\text{g}}}$= - 0.5 m/s
The negative sign indicates that the rifle recoils backwards with a velocity of 0.5 m/s
$ \Rightarrow $ $\left| {{{\text{v}}_{\text{g}}}} \right|$=0.5 m/s
Therefore, 0.5 m/s is the recoil velocity of the gun of mass 8 kg.
Hence, option (D) is the correct answer.

Note- As we know, linear momentum is the product of the mass and velocity of an object. And also, it is a vector quantity which possesses a magnitude and a direction. If an object's having mass m and velocity v is given, then the object's momentum is defined as:
${\text{P = mv}}$.