Question

For a fixed initial speed, the range of a projectile is determined by the angle at which it is fired. For all but the maximum, there are two angles that give the same range. Considering factors that might affect the ability of an archer to hit a target, such as wind, explain why the smaller angle (closer to the horizontal) is preferable. When would it be necessary for the archer to use the larger angle? Why does the punter in a football game use the higher trajectory?

Short answer

Because it takes less time to hit the target, a small angle is preferred. The archer uses a larger angle when the initial speed is fixed, and the target is far away, but air resistance and wind affect the ability to strike the target. The punter wants the football to take the highest possible trajectory, in order to have the longest possible flight time while covering the greatest horizontal distance.

Step-by-step solution

Projectile motion

When a projectile is thrown in the air at an angle to the horizontal, it follows a curved route known as the trajectory due to gravity's effect, and this motion is known as projectile motion.

The time flight and the horizontal range of the projectile are determined by its initial velocity and the angle of projection from the horizontal.

Why smaller angle is preferable

When a projectile is projected with an initial velocity at some angle from the horizontal, the time of flight of the projectile is

$t=\frac{2u\sin \theta }{g}$ (1.1)

Here g is the acceleration due to gravity.

From equation (1.1), it is clear that for a small angle of projection, the time of flight will be short. So, the arrow will suffer the small effect of wind and air resistance and hit the target more precisely.

Hence, the small-angle is preferable because it takes a shorter time to hit the target.

Why archers use large angles

The range of the projectile projected with an initial velocity u at some angle$\theta$ from the horizontal is

$R=\frac{u^2\sin \left(2\theta \right)}{g}$ (1.2)

When the initial speed is fixed and the target is far, the archer uses a larger angle to hit the target. But as the angle of projection increases, the time of flight increases, which leads to the effect of wind and air resistance for a longer time, and the chances that the target will not be hit are more.

Punter uses higher trajectory

In order to achieve a higher trajectory, the angle of projection must be high.

From equation (1.1), it is clear that for a larger angle (higher trajectory), the time of flight will be longer than a shorter angle. It will give the team members enough time so that they can get downfield to make the return stop.

From equation (1.2), it is clear that for fixed initial speed, the range of the projectile depends on the angle of projection; the larger the angle (higher trajectory), the greater will be the range. In the case of a football game, the requirement of the punter is to kick the ball as far down as possible.

Hence, the punter needs that the football will take the highest trajectory possible to have a longer time of flight while covering the longest horizontal distance.