Find the velocity, acceleration and rate of displacement

Question

A particle moves in such a way that its displacement, x cm, t seconds after starting to move from a fixed point O is given by x(t)=10t +20e-0.2t, t≥0

i) Determine the rate if change of its displacement at time t=5.

ii) Will this particle ever maintain a constant velocity for a period of time?

iii) Will this particle ever maintain a constant acceleration for a period of time?

Answer 1

To find the rate of change (velocity), take the derivative of x(t) wrt t:

 

Velocity = dx/dt = 10 - 4e^-0.2t

 

i)  At t = 5:

 

Velocity = 10 - 4e^-0.2*5 = 10 - (4/e) ≅ 8.53 cm/sec

 

ii)  Due to the exponential term, the velocity is never constant but for large values of t, dx/dt → 10 cm/sec

 

iii)  Acceleration is the second derivative of x(t):

 

Acceleration = d²x(t)/dt² = 0.8e^-0.2t

 

Due to the exponential term, the acceleration is never constant; but for large values of t, d²x/dt² → 0