A basketball player who is 2.00 m tall is standing on the...

A basketball player who is $2.00 \mathrm{~m}$ tall is standing on the floor $10.0 \mathrm{~m}$ from the basket, as in Figure $\mathrm{P} 4.52 .$ If he shoots the ball at a $40.0^{\circ}$ angle with the horizontal. at what initial speed must he throw so that it goes through the hoop without striking the backboard? The basket height is $8.05 \mathrm{~m}$.

Key Concepts

Time of Flight Analysis

Determining the time of flight involves analyzing the vertical motion of the projectile, particularly by solving the kinematic equations for the time needed to reach a specified vertical displacement. This time then links to the horizontal distance traveled, enabling the solution of target hitting problems.

Equations of Motion

The equations of motion for uniformly accelerated systems describe the relationships among displacement, velocity, acceleration, and time. For projectile motion, these equations allow one to compute the horizontal distance traveled and the vertical displacement at any point in time, taking gravitational acceleration into account.

Relative Height Consideration

In scenarios involving different initial and final heights, such as a shooter’s release point and a target at a different elevation, it is essential to incorporate the vertical displacement between these points. This adjustment ensures that the projectile trajectory intersects the target at the correct height.

Projectile Motion

This concept describes the motion of an object under the influence of gravity alone, following a parabolic trajectory when air resistance is negligible. The analysis involves separating the motion into horizontal and vertical components, where the horizontal motion proceeds at constant velocity and the vertical motion is uniformly accelerated due to gravity.

Decomposition of Initial Velocity

In projectile problems, the initial speed is split into horizontal and vertical components using trigonometric functions such as sine and cosine relative to the launch angle. This decomposition is crucial for independently analyzing the time evolution of the projectile in each direction.

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