Chapter 6, Problem 6STP

To determine

To Choose: The correct option.

Answer to Problem 6STP

Option B. $16\text{m}$

Explanation of Solution

Given:

An apple is shot from a slingshot at an angle of $78°$ from the horizontal with an initial velocity of 18m/s.

Formula used:

Final velocity with constant acceleration can be obtained by first equation of motion:

  $v_f=v_i+at$

Where, $v_f$ is the final velocity, $v_i$ is the initial velocity and $t$ is the time taken.

The displacement of an object with a constant acceleration can be obtained by third equation of motion:

  $d_f=v_{i}t+\frac{1}{2}a{t}^2$

Where, $d_f$ is displacement, $v_i$ is the initial velocity, a is the acceleration and $t$ is the time taken

The rectangular components of a vector A making an angle of $\theta$ with the horizontal axis or x -axis, is given by,

  $A_x=A\cos \theta$

  $A_y=A\sin \theta$

Calculation:

The components of the initial velocity in the vertical and horizontal direction is,

  $\begin{array}{l}{v}_{ix}=v_i\cos 78°=18\cos 78°=3.74\text{m/s}\\ v_{iy}=v_i\sin 78°=18\sin 78°=17.60\text{m/s}\end{array}$

Acceleration acting on the apple is in the vertical direction which is due to gravity only, hence it can be written as

  $\begin{array}{l}{a}_x=0\mathrm{m}/s^2\\ a_y=-9.8\mathrm{m}/s^2\end{array}$

Negative sign shows that this acceleration acts in opposite direction (downward direction) to the motion of the apple.

Also, it is known that the ball comes to momentarily rest at the maximum height, thus at maximum height,

  $v_{fy}=0\text{m/s}$

Now, considering the vertical motion of the apple to determine the time taken to attain maximum height as,

  $\begin{array}{c}{v}_{fy}=v_{iy}+a_{y}t\\ 0=17.60+\left(-9.8\right)t\\ t=1.79\text{s}\end{array}$

Then, the maximum height attained by the apple will be,

  $\begin{array}{c}{d}_{\max }=v_{iy}t+\frac{1}{2}{a}_y{t}^2\\ =\left(17.60\right)\left(1.79\right)+\frac{1}{2}\left(-9.8\right){\left(1.79\right)}^2\\ =31.504-15.700\\ =15.804\text{m}\\ \approx \text{16}\text{.0 m}\end{array}$

Conclusion:

Thus, the correct option is B.