A ball is to be shot from level ground with a certain speed. Figure 4-45 shows the range R it will have versus the launch angle u0. The value of u0 determines the flight time; let tmax represent the maximum flight time. What is the least speed the ball will have during its flight if u0 is chosen such that the flight time is 0.500tmax?
The Correct Answer and Explanation is :
In projectile motion, the flight time ( T ) of a projectile launched from and landing on level ground is given by:
[ T = \frac{2 v_0 \sin \theta_0}{g} ]
where ( v_0 ) is the initial launch speed, ( \theta_0 ) is the launch angle, and ( g ) is the acceleration due to gravity. The maximum flight time, ( T_{\text{max}} ), occurs at a launch angle of 90°, resulting in:
[ T_{\text{max}} = \frac{2 v_0}{g} ]
Given that the desired flight time is half of the maximum, we have:
[ T = 0.5 \times T_{\text{max}} = \frac{v_0}{g} ]
Equating the two expressions for ( T ):
[ \frac{2 v_0 \sin \theta_0}{g} = \frac{v_0}{g} ]
Simplifying, we find:
[ 2 \sin \theta_0 = 1 ]
[ \sin \theta_0 = \frac{1}{2} ]
Therefore:
[ \theta_0 = \sin^{-1}\left(\frac{1}{2}\right) = 30^\circ ]
The horizontal and vertical components of the velocity at launch are:
[ v_{0x} = v_0 \cos \theta_0 = v_0 \cos 30^\circ = v_0 \times \frac{\sqrt{3}}{2} ]
[ v_{0y} = v_0 \sin \theta_0 = v_0 \sin 30^\circ = v_0 \times \frac{1}{2} ]
At the peak of its trajectory, the vertical component of the velocity becomes zero, leaving only the horizontal component. Thus, the minimum speed of the ball during its flight is:
[ v_{\text{min}} = v_{0x} = v_0 \times \frac{\sqrt{3}}{2} ]
This result indicates that when the flight time is half of the maximum possible, the ball reaches its minimum speed at the peak of its trajectory, which is ( \frac{\sqrt{3}}{2} ) (approximately 0.866) times the initial launch speed.
This analysis demonstrates the relationship between launch angle, flight time, and speed in projectile motion. By selecting a launch angle of 30°, the projectile achieves a flight time that is half of the maximum, and its minimum speed during flight occurs at the apex, determined solely by the horizontal component of the initial velocity.