Rates of Change

Question

An object is moving in a straight line from a fixed point. The displacement s(in metres) is given by s=-2t^2+28t+45, t is greater than or equal to 0, where t is in seconds. 

 

Find t when the object reaches its maximum displacement.

 

Find the maximum displacement reached by this object. 

 

Determine the acceleration at any time 

Answer 1

Maximum displacement is when ds/dt = 0.

 

ds/dt = -4t + 28 = 0 ⇒ t = 7s

 

s_max = s(7) = -2(7)² + 28(7) + 45 = 143 m

 

a(t) = d²s/dt² = -4 m/s²

 

Answer 2

Finding the extrema by setting the derivative equal to zero and solving for x:

0 =s'(x) = -4t + 28

 

Extrema occurs at t = 7.

s(7) = -2(7)^2 + 28*7 + 25 = -98 + 196 + 45 = 143

 

Accelaration is the second derivative:

 

s''(x) = -4