Question

Sally is driving along a straight highway. At t = 0, when she is moving in the +x direction at 10 m/s, she passes a signpost at x = 50 m. Her acceleration as a function of time is ax = 2.0 m/s2 – (0.10 m/s3)t . (a) Find her velocity and position x as a function of time. (b) When is her velocity greatest? (c) What is the maximum velocity? (d) Where is the car when it reaches that maximum velocity?

Answer 1

given

ax = 2 - 0.1*t

a) ax = dvx/dt

dvx = ax*dt

integral dvx = integral ax*dt

vx - vox = integral (2 - 0.1*t)*dt

vx - 10 = 2*t - 0.1*t^2/2

vx = 10 + 2*t - 0.05*t^2 <<<<<<<-----------Answer

b) when velocity is maximum,

dvx/dt = 0

0 + 2 - 0.05*2*t = 0

t = 2/(0.05*2)

= 20 s <<<<<<<-----------Answer

c) Vmax = 10 + 2*20 - 0.05*20^2

= 30 m/s <<<<<<<-----------Answer

d) vx = dx/dt

==> dx = vx*dt

integral dx = integral vx*dt

x - xo = integral (10 + 2*t - 0.05*t^2)*dt

x - 50 = 10*t + 2*t^2/2 - 0.05*t^3/3

x = 50 + 10*t + 2*t^2/2 - 0.05*t^3/3

x_max = 50 + 10*30 + 2*30^2/2 - 0.05*30^3/3

= 800 m