Question

a projectile lands at the same height from which it was launched. which initial velocity will result in the greatest horizontal displacement of the projectile? neglect friction.
2018
5.
a soccer ball is kicked into the air from level ground with an initial speed of 20. meters per second and returns to ground level. at which angle above the horizontal should the ball be kicked in order for the ball to travel the greatest total horizontal distance? neglect friction.
(1) ( 15^{circ} )
(2) ( 30.^{circ} )
( 45^{circ} )
( 75^{circ} )
2017
37.

Explanation:

Step1: Recall the range formula for projectile motion

The range formula for projectile motion is \( R=\frac{v_{i}^{2}\sin2\theta}{g} \), where \( v_{i} \) is the initial velocity, \( \theta \) is the launch angle, and \( g \) is the acceleration due to gravity. Since \( v_{i} \) and \( g \) are constant in this problem ( \( v_{i} = 20\space m/s\), \( g= 9.8\space m/s^{2}\)), we need to maximize \( \sin2\theta \).

Step2: Find the value of \( \theta \) that maximizes \( \sin2\theta \)

The sine function \( y = \sin x \) has a maximum value of \( 1 \). We set \( 2\theta=\sin^{- 1}(1)\). Solving for \( \theta \), when \( \sin2\theta = 1\), \( 2\theta=90^{\circ}\), so \( \theta = 45^{\circ}\).

Answer:

\(45^{\circ}\) (Option (3) if we assume the options are labeled as (1) \(15^{\circ}\), (2) \(30^{\circ}\), (3) \(45^{\circ}\), (4) \(75^{\circ}\))