Question

Velocity of a Particle The displacement s (in meters) of a particle moving in a straight line is given by the equation of motion $s=4t^3+6t+2$, where t is measured in seconds. Find the instantaneous velocity of the particle s at times

$t=a,t=1,t=2,t=3$.

Short answer

The instantaneous velocities are $12a^2+6$, 18, 54, 114 respectively.

Step-by-step solution

Step 1.  Given

The displacement s (in meters) of a particle moving in a straight line is given by the equation of motion $s=4t^3+6t+2$, where t is measured in seconds.

Step 2.  To determine

Find the instantaneous velocity of the particle s at times

$t=a,t=1,t=2,t=3$.

Step 3.  Calculation

To find the instantaneous velocity of the particle we need to find the first derivative of s.

$\begin{array}{l}s=4t^3+6t+2\\ s\text{'}=4\left(3t^2\right)+6\left(1\right)+0\\ s\text{'}=12t^2+6\end{array}$

At $t=a,$$s\text{'}=12a^2+6$

At$t=1,$$s\text{'}=12{\left(1\right)}^2+6=12+6=18$

At$t=2,$$s\text{'}=12{\left(2\right)}^2+6=48+6=54$

At $t=3$$s\text{'}=12{\left(3\right)}^2+6=108+6=114$

So, the instantaneous velocities are $12a^2+6$, 18, 54, 114 respectively.