Chapter 6, Problem 75A

To determine

Whether the height that one can reach during the long jump improve performance. The effect on the length of the jump.

Explanation of Solution

Introduction:

A type of motion in which object moves along curved path under the action of gravity.The horizontal component of velocity remains constant because there is no acceleration acting in the horizontal direction. In the vertical direction, acceleration due to gravity acts and so, vertical motion is accelerated.

The range of a projectile motion can be expressed as:

$R=\frac{u^2\sin 2\theta }{g}$

Here, u is the initial velocity, g is the acceleration due to gravity and $\theta$ is the angle of the projectile.

The maximum height achieved can be expressed as:

$H=\frac{u^2{\sin }^2\theta }{2g}$

Here, u is the initial velocity, g is the acceleration due to gravity and $\theta$ is the angle of the projectile.

Both height and range of a projectile depend on the initial velocity and angle of projectile. The range is maximum when angle of projectile is $45°$ and maximum height is reached when angle of projectile is $90°$ .

Take the ratio:

$\begin{array}{c}\frac{R}{H}=\frac{2\sin 2\theta }{{\sin }^2\theta }\\ =\frac{4\sin \theta \cos \theta }{{\sin }^2\theta }\\ =4\cot \theta \\ R=4H\cot \theta \\ R_{\max }=4H\end{array}$

For maximum range, the projectile angle should be $45°$ and the length of the jump would be 4 times the maximum height achieved.