Chapter 3.4, Problem 19E

To determine

To calculate: Find (a) the displacement during first five second.

(b) The average velocity during the first five second.

(c) The instantaneous velocity when $t=4\sec$

(d) The acceleration of the particle when $t=4\sec$.

(e) The time t when the particle changes the direction.

(f) The time t when s is a minimum.

Answer to Problem 19E

The correct answer is (a) $s=12m$

(b) $v=2.4m/\sec$

(c) $V_{inst.}=5m/\sec$.

(d) $a=2m/{\sec }^2$

(e) $t=1.5\sec$

(f) $t=1.5\sec$

Explanation of Solution

Given information: In the question, it is given that a particle moves along a line and distance function is also given $s=t^2-3t+2$.

Formula used: We know that, Instantaneous velocity is $V(t)=\frac{ds}{dt}$ .Instantaneous acceleration is $a(t)=\frac{d^{2}s}{d^{2}t}$ .

Calculation:

1. For finding displacement at first five second we put the value of $t=5$ in given displacement function.

  $\begin{array}{l}s=t^2-3t+2\\ s(5)=25-15+2=12\end{array}$

(b) For finding average velocity.

  $Averagevelocity=\frac{totaldisplacement}{totaltime}$

  $Averagevelocity=\frac{s(5)}{5}=\frac{12}{5}=2.4m$

(c) Instantaneous velocity is $V(t)=\frac{ds}{dt}$.

  $\begin{array}{l}s=t^2-3t+2\\ \frac{ds}{dt}=2t-3\end{array}$

Instantaneous velocity at $t=4\sec$

  $\begin{array}{l}\frac{ds}{dt}=2t-3\\ V_{inst.}=5m/\sec \end{array}$

(d) For Instantaneous acceleration is $a(t)=\frac{d^{2}s}{d^{2}t}$ .

  $\begin{array}{l}s=t^2-3t+2\\ \frac{ds}{dt}=2t-3\\ a=\frac{d^{2}s}{d{t}^2}=2m/{\sec }^2\end{array}$

(e) The value of t where the particle change direction when $\frac{ds}{dt}=0$

  $\begin{array}{l}s=t^2-3t+2\\ \frac{ds}{dt}=2t-3=0\\ t=\frac{3}{2}=1.5\sec \end{array}$

(f) For finding $s_{\min }$ we have to find vertex of this quadratic equation.

  $\begin{array}{l}s=t^2-3t+2\\ $ s={(t-3/2)}^2-1/4$\end{array}$

From here I can say that, where $s_{\min }$ is means $t=3/2\sec$.

Thus, the correct answer is (a) $s=12m$

(b) $v=2.4m/\sec$

(c) $V_{inst.}=5m/\sec$.

(d) $a=2m/{\sec }^2$

(e) $t=1.5\sec$

(f) $t=1.5\sec$