A car of mass m accelerates from rest along a level straight track, not at const

Question

A car of mass m accelerates from rest along a level straight track, not at constant acceleration but with constant engine power, P. Assume that air resistance is negligible.

(a) Find the car's velocity as a function of time.

(b) A second car starts from rest alongside the first car on the same track, but maintains constant acceleration. Which car takes the initial lead? Does the other car overtake it? If yes, write a formula for the distance from the starting point at which this happens.

(c) You are in a drag race, on a straight level track, with an opponent whose car maintains a constant acceleration of 12 m/s2. Both cars have identical masses of 1000 kg. The cars start together from rest. Air resistance is assumed to be negligible. Calculate the minimum power your engine needs for you to win the race, assuming the power output is constant and the distance to finish line is 0.250 mi.

Explanation / Answer

a) the car's kinetic energy K=P*t. where P is the engine power. K=mv^2/2 so v=sqrt(2K/m)=sqrt(2Pt/m). b)the velocity of the 2nd car, v'=a*t. so s=at^2/2. --- distance traveled by the 1st car. v=ds/dt so v*dt=ds. sqrt(2Pt/m)*dt=ds. sqrt(2P/m) *t^3/2=S. -------- initially, acceleration rate by the 1st car. a=v'=sqrt(P/2mt). so at t=0, a become very big, so the 1st car will over come, but then since at t large, t^2>t^3/2. so the 2nd car will be faster, distance of meating. sqrt(2P/m)*t^3/2=at^2/2 2sqrt(2P/m)=a*t^1/2 4*2P/m=a^2*t. so t=8P/a^2m so distance. S=at^2/2=a*64P^2/2a^4m^2=32P^2/a^3m^2. ---------- c). we have that S=32P^2/a^3m^2 402=32*P^2/12^3*1000^2 so P=1.5e5(W)

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