Consider a muscle car that is designed for maximum acceleration. The static coef

Question

Consider a muscle car that is designed for maximum acceleration. The static coefficient of friction between tires and ground is µs=1.1. The vehicle has a mass of 2500 kg and the motor can do work at a maximum rate of P=250 kW.

(a) Determine the maximum acceleration possible and the time to reach 60 mph if there were no limitation on the power available from the engine.

(b) Determine the minimum time needed to reach 60 mph neglecting any limitations on instantaneous acceleration due to the static coefficient of friction.

(c) In reality acceleration is initially limited by µs and then by the power available from the motor. Beyond what speed is acceleration limited by the power availability?

(d) Determine the minimum time needed to reach 60 mph considering both µs and P. (Hint: Initially there will be motion with constant acceleration. Followed by motion subject to a constant power input.)

Explanation / Answer

a) acceleration will be provided by friction force.

Maximum static friction = us g = 1.1 x 9.81 = 10.79 m/s^2

b) v = 60 mph = 60 x 1609 / 3600 = 26.82 m/s

using v = u + at

26.82 = 0 + 10.79t

t = 2.49 sec

c) P = Fv

250 x 10^3 = (usMg)v

v = 250 x 10^3 / (1.1 x 2500 x 9.81) = 9.27 m/s

so we found the max. speed that can car attain still using max. friction.
after acc. will get on decrease as speed increase.

d) after 9.26 m/s .

after 9.26 ms/ ...... acc. = dv/dt

force will be F = mdv/dt

and speed is v.

so Power = (m dv/dt) (v)

mvdv = Pdt

2500 vdv = 250 x 10^3 dt

intergating both sides.,

[v^2 /2 ] = 100t

(26.82^2) - (9.26^2) = 200( t - 0 )

t =3.16 sec

time taken to reach 0 to 9.26 sec   t = 9.26 / 10.79 = 0.86 sec

total time = 3.16 + 0.86 = 4.02 sec

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