Chapter 2, Problem 36P

(a)

To determine

The velocity of the object as a function of time.

Answer to Problem 36P

The velocity of the object as a function of time is $v\left(t\right)=\left(5.0{\text{m/s}}^2\right)tfor0<t<10\text{s}$ and $v(t)=\left(15.0{\text{m/s}}^{\text{2}}\right)t-100\text{m/s }for10<t<10\text{s}$.

Explanation of Solution

Write the equation for the acceleration over a given time.

    $\begin{array}{c}a=\frac{\Delta v}{\Delta t}\\ \Delta v=a\Delta t\end{array}$        (I)

Here, $a$ is the acceleration, $\Delta v$ is the change in velocity over the change in time period $\Delta t$.

Conclusion:

Substitute $5.0{\text{m/s}}^{\text{2}}$ for $a$ and $t-0\text{s}$ for $\Delta t$ in equation (I).

    $\begin{array}{c}\Delta =v(t)-v\left(0\right)\\ =\left(5.0{\text{m/s}}^{\text{2}}\right)\left(t-0\text{s}\right)\\ =\left(5.0{\text{m/s}}^{\text{2}}\right)t\end{array}$        (II)

The velocity at $t=0$ , $v(0)$ is zero. Hence rewrite equation (II).

    $v\left(t\right)=\left(5.0{\text{m/s}}^2\right)tfor0<t<10\text{s}$        (III)

It is a constant acceleration of $15{\text{m/s}}^{\text{2}}$ from $t=10\text{s}$ to $t=26\text{s}$.

    $\begin{array}{c}\Delta v=v(t)-v\left(10\text{s}\right)\\ =\left(15{\text{m/s}}^{\text{2}}\right)\left(t-10\text{s}\right)\\ =\left(15.0{\text{m/s}}^{\text{2}}\right)t-150\text{m/s}\end{array}$

Hence,

    $v(10\text{s) = 50m/s}$

Therefore, write the velocity of the object as a function of time.

    $v(t)=\left(15.0{\text{m/s}}^{\text{2}}\right)t-100\text{m/s }for10<t<10\text{s}$        (VI)

(b)

To determine

The velocities of the object.

Answer to Problem 36P

The new velocities of the object are $v\left(t\right)=\left(5.0{\text{m/s}}^2\right)t+40\text{m/s}for0<t<10\text{s}$ and $v(t)=\left(15.0{\text{m/s}}^{\text{2}}\right)t-60\text{m/s }for10<t<10\text{s}$.

Explanation of Solution

The initial velocity here is $40\text{m/s}$. Therefore, adding $40\text{m/s}$ to equation (III) and equation (IV) gives the answers.

Conclusion:

Therefore, the new velocities are

    $v\left(t\right)=\left(5.0{\text{m/s}}^2\right)t+40\text{m/s}for0<t<10\text{s}$

And

    $v(t)=\left(15.0{\text{m/s}}^{\text{2}}\right)t-60\text{m/s }for10<t<10\text{s}$