Question

A laser velocimeter tracks the speed of a car using the Doppler effect for light. When a velocimeter tracks a car and the results are analyzed by a computer givng a best fit function:

v(t) = (2.1 m/s²) t - (2e-06 m/s⁶) t⁵

as it accelerates from rest to its cruising speed.

(a) How long does it take for the car to reach its cruising speed?


(b) What is its cruising speed?


(c) How far has the car traveled when it reaches its cruising speed?

Answer 1

Given that,

v = 2.1*t - (2*10^(-6)*t^5)

acceleration, a = dv / dt = d(2.1*t - (2*10^(-6)*t^5)) / dt

a = 2.1 - (10*10^(-6)*t^4)

for the car to reach its cruising speed,

a = 0

2.1 - (10*10^(-6)*t^4) = 0

t = 21.4 s

Time taken  for the car to reach its cruising speed,

t = 21.4 s

(b)

cruising speed,

v = 2.1*21.4 - (2*10^(-6) * 21.4⁵)

v = 35.9 m/s

(c)

distance  traveled when car reaches its cruising speed,

d = [2.1*t - (2*10^(-6)*t^5)] . dt

d = 1.05*t^2 - (0.33*10^(-6)*t^6)

d = 1.05*21.4² - (0.33*10^(-6) * 21.4^6)

d = 432.8 m