Can someone please help me with my Physics Question?

Question

A woman fires a rifle with barrel length of 0.6600 m. Let (0, 0) be where the 75 g bullet begins to move, and the bullet travels in the +x-direction. The force exerted by the expanding gas on the bullet is (16,000 + 10,000x − 24,000x²) N, where x is in meters.

(a) Calculate the work done (in kJ) by the gas on the bullet as the bullet travels the length of the barrel. (Enter your answer to at least two decimal places.)

______ kJ

(b) If the barrel is 1.040 m long, how much work (in kJ) is done? (Enter your answer to at least two decimal places.)

 ______kJ

(c) How does this value compare with the work calculated in part (a)?

The work is greater by  _____%.

Answer 1

Work is Force times displacement cosine angle between them.

Angle = 0 deg so cosine = 1

Since Force is changing with displacement one must Integrate

a / I = Integral F dx evaluated from 0 to 0.66

I = 16,000 x + 5000x^2 − 8,000x^3

I = 16,000 (0.66) + 5000 (0.66)^2 − 8,000 (0.66)^3 - 0

I = 10560 + 2178 - 2300 = 10438

b /

I = 16,000 (1.04) + 5000 (1.04)^2 − 8,000 (1.04)^3

I = 16640 + 5408 -- 8999 = 13049

c / The percentage increase due to longer barrel length

(13049 - 10438 ) / 10438 = 25%