Chapter 4.8, Problem 44E

To determine

To show: The displacement of a straight line after time t is $s=\frac{1}{2}a{t}^2+v_{0}t+s_0$.

Explanation of Solution

Given:

Acceleration is a, initial velocity is $v_0$, and initial displacement is $s_0$.

Formula used:

The expression for acceleration function $a\left(t\right)$.

$a\left(t\right)=v^{\prime }\left(t\right)$ (1)

Here,

$v^{\prime }\left(t\right)$ is first derivative of velocity function $v\left(t\right)$.

The expression for velocity function $v\left(t\right)$.

$v\left(t\right)=s^{\prime }\left(t\right)$ (2)

Here,

$s^{\prime }\left(t\right)$ is first derivative of displacement function $s\left(t\right)$.

Antiderivative of $t$ is $\frac{1}{2}{t}^2$ and 1 is $t$.

Calculation:

Initial velocity is $v_0$, that is $v\left(0\right)=v_0$ and the initial displacement is $s_0$, that is $s\left(0\right)=s_0$.

Given $a\left(t\right)=a$.

Substitute $a$ for $a\left(t\right)$ in equation (1),

$v^{\prime }\left(t\right)=a$

Integrate the equation $v^{\prime }\left(t\right)=a$ with respect to t as follows.

$v\left(t\right)=at+C$ (3)

Where C is an arbitrary constant.

Substitute 0 for t in equations (3),

$\begin{array}{c}v\left(0\right)=a\left(0\right)+C\\ =C\end{array}$

Substitute $v_0$ for $v\left(0\right)$, $C=v_0$.

Substitute $v_0$ for C in equation (3), $v\left(t\right)=at+v_0$.

Substitute $at+v_0$ for $v\left(t\right)$ in equation (2), $s^{\prime }\left(t\right)=at+v_0$.

Obtain the antiderivate of $s^{\prime }\left(t\right)=at+v_0$ with respect to t as follows.

$s\left(t\right)=a\left(\frac{1}{2}{t}^2\right)+v_{0}t+D$

$s\left(t\right)=\frac{1}{2}a{t}^2+v_{0}t+D$ (4)

Where D is an arbitrary constant.

Substitute 0 for t in equation (4),

$\begin{array}{c}s\left(0\right)=\frac{1}{2}a\left(0^2\right)+v_0\left(0\right)+D\\ =0+D\\ =D\end{array}$

Substitute $s_0$ for $s\left(0\right)$, $D=s_0$.

Substitute $s_0$ for D in equation (4), $s\left(t\right)=\frac{1}{2}a{t}^2+v_{0}t+s_0$.

Thus, the displacement of a straight line after time t is $\underset{\_}{s=\frac{1}{2}a{t}^2+v_{0}t+s_0}$.