# Analysis of a bullet undergoing constant acceleration

A gun fires a bullet of mass 22 grams out of a barrel 36 cm long. The gun is attached to a spring. From the recoil of the spring and the masses of the gun and the spring we determine that the gun recoiled with a total momentum of 4.7 kg m/s.

- With what velocity did the bullet exit the barrel?

- Assuming that the bullet accelerated uniformly from rest along the length of the barrel, how long did it take the bullet to accelerate from rest down the length of the barrel?

- What was the average force exerted on the bullet as it accelerated along the length of the barrel?

- What average force would be felt by the individual holding the gun for the time the bullet accelerates along the length of the barrel?

https://brainmass.com/physics/acceleration/analysis-bullet-undergoing-constant-acceleration-395816

## SOLUTION This solution is **FREE** courtesy of BrainMass!

- Conservation of linear momentum applies

Let momentum of Gun/Spring assembly p(g) = 4.7 kg.m/s

Momentum of bullet p(b) is the product of its exit velocity v(b) and mass m(b) = 22g = 0.022 kg

p(b) = m(b)*v(b) = p(g)

v(b) = p(g)/m(b) = 4.7/0.022 = 213.64 m/s

- For linear acceleration of the bullet use one of the equations from linear kinematics for constant acceleration (a) in terms of final velocity v(b), initial velocity u(b) = 0 (starts from rest), time (t), and the distance traversed along the gun barrel s = 36cm = 0.36m

v(b)^2 = u(b)^2 + 2a*s

a = {v(b)^2 - u(b)^2}/2s u(b) = 0

a = 213.64^2/2*0.36 = 63,389.6 m/s2

Now use the equation for linear constant acceleration as

a = {final velocity - initial velocity}/time

a = {v(b) - u(b)}/t

t = {v(b) - u(b)}/a = {213.64 - 0}/63,389.6

t = 3.37 ms

- Average force on bullet as it accelerates under constant acceleration (a) is given by Newton's second law

F = m*a = 0.022 x 63,389.6 = 1394.57 N

- Force felt by someone holding the gun would be the reaction force which would be 1394.57 N but acting in the opposite direction to the force on the bullet

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