Question

A ball is shot from ground level over level ground at a certain initial speed. Figure 4-27 gives the range R of the ball versus its launch angle u0. Rank the three lettered points on the plot according to (a) the total flight time of the ball and (b) the ball’s speed at maximum height, greatest first.

Short answer

1. $\mathrm{Rank}:c>b>a$
   

1. $\mathrm{Rank}:a>b>c$
   

Step-by-step solution

Given information

Curve of Range vs initial angle of projection

To understand the concept

It deals with the kinematic equations of motion in which the motion can be described with constant acceleration. Also, this problem is based on the concept of projectile motion. When a particle is thrown near the earth’s surface, it travels along a curved path under constant acceleration directed towards the center of the earth surface. This is a projectile motion problem, which contains two independent motions.

Horizontal motion with constant speed ${\mathbf{a}}_{\mathbf{x}}\mathbf{=}\mathbf{0}$

Vertical motion with acceleration due to gravity $\mathbf{a}\mathbf{=}\mathbf{9}\mathbf{.}\mathbf{8}\mathbf{}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$.

Formulae:

$$\begin{gathered} \mathrm{Range}\left(R\right)=v_{\mathit{0}x}T \\ {v}_{0x}=\mathrm{Horizontal}\mathrm{component}\mathrm{of}\mathrm{initial}\mathrm{speed} \\ T-\mathrm{Time}\mathrm{of}\mathrm{flight} \end{gathered}$$

(a) To rank the three lettered points on the plot according to the total time of flight of the ball

We know that,

$\begin{array}{l}R=V_0{x}^T\\ =V_0\times \cos \left({\theta }_0\right)\times T\end{array}$

Hence,

$T=\frac{R}{V_0\cos \left({\theta }_0\right)}$

As$\theta$increases, time of flight also increases, value of$cos\left(\theta \right)$would decrease. Therefore, rank according to time is:

$\mathrm{Rank}:c\mathit{>}b\mathit{>}a$

(b) To rank the three lettered points on the plot according to the ball’s speed at maximum height, greatest first

At maximum height vertical component of velocity is zero, so speed is same as horizontal component of velocity which is same at any point.

Hence,

$\begin{array}{l}v=V_{0}x\\ =V_0\times \cos \left({\theta }_0\right)\end{array}$

As$\theta$increases, speed decreases. Therefore, rank according to speed at maximum height is:

$\mathrm{Rank}:a>b>c$