8. (10 points) Stopping Distance. A car is traveling on a level road with speed

Question

8. (10 points) Stopping Distance. A car is traveling on a level road with speed v_0 at the instant when the brakes lock, so that the tires slide rather than roll. (a) Use the work- energy theorem to calculate the minimum stopping distance of the car in terms of v_0,g, and the coefficient of kinetic friction mu_k between the tires and the road. (b) By what factor would the minimum stopping distance change if (i) the coefficient of kinetic friction were doubled, or (ii) the initial speed were doubled, or (ii) both the coefficient of kinetic friction and the initial speed were doubled?

Explanation / Answer

(a)

By work energy theorem,

W = change in KE

FS = change in KE = ½ * m*v0^2

Where F is the frictional force = u_k*m*g

S is the stopping distance

So, rearranging the eqn:

S = mv0^2/2*u_k*m*g

=> S = v0^2/(2*u_k*g)

(b)

(i)Now,

S’ = v0^2/(2*2u_k*g)

So

S’/S = 1/2

(ii) Here,

S’ = (2v0)^2/(2*u)k*g) = 4S

SO,

S’/S = 4

(iii)

Both conditions apply here, so simply multiply results of (i) and (ii)

S’/S = 4*1/2 = 2

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