Question

A 149 g bullet is fired from a rifle having a barrel 0.504 m long. Assuming the origin is placed where the bullet begins to move, the force exerted on the bullet by the expanding gas is f = a b x - c x2, where a = 8130 N, b = 12300 N/m, c = 37800 N/m2, with x in meters. Determine the work done by the gas on the bullet as the bullet travels the length of the barrel. Answer in units of J.

Answer 1

The work done on the bullet by the expanding gases as it travels the length of a 0.504 m rifle barrel is approximately 81539 Joules.

Step-by-step explanation:

To determine the work done by the gas on the bullet as it travels the length of the rifle barrel, we integrate the force function over the distance the bullet travels. Given the force function f = a × b × x - c × x2, with constants a = 8130 N, b = 12300 N/m, and c = 37800 N/m2, the work done is the integral of this function from x = 0 to x = 0.504 m (the length of the barrel). The work done (W) can be calculated as W = ∫ (abx - cx2) dx from 0 to 0.504, W = ab ∫ x dx from 0 to 0.504 - c ∫ x2 dx from 0 to 0.504, W = ab [x2/2] from 0 to 0.504 - c [x3/3] from 0 to 0.504, W = ab [0.5042/2] - c [0.5043/3], W = (8130 × 12300 × 0.5042/2) - (37800 × 0.5043/3), W = (8130 × 12300 × 0.254016/2) - (37800 × 0.128024064/3), W = 83156799.6 J - 1617296.271 J, W = 81539493.329 J. Therefore, the work done on the bullet by the expanding gases is approximately 81539 Joules (J) when it travels the entire length of the barrel.