Question

The initial velocity v (in feet per second) at which a person must jump to make a slam dunk is given by R = 21/2 - 1/64v^2, where R is the person's standing reach in feet.

Answer 1

The initial velocity (v) at which a person must jump to make a slam dunk is given by the equation
`\(R = √(2) - (1)/(64)v^2\), where \(R\)` is the person's standing reach in feet.

Step-by-step explanation:

The given equation
`\(R = √(2) - (1)/(64)v^2\)` relates the standing reach (R) of a person to the initial velocity (v) required for a successful slam dunk. To understand this equation, we can break down its components.

The term
`\(√(2)\)` represents the initial standing reach without the need for any additional jump velocity. The second term
`\((1)/(64)v^2\)` is the contribution from the jump velocity, which is squared. This term implies that the jump velocity has a quadratic relationship with the change in standing reach.

To solve for (v), one can rearrange the equation by isolating the (v) term:

`\[ v^2 = 64(√(2) - R) \]`

`\[ v = \sqrt{64(√(2) - R)} \]`

This expression gives the initial velocity required for a slam dunk based on the person's standing reach. It is essential to note that the value inside the square root must be non-negative for a physically meaningful solution.

In conclusion, the equation provides a mathematical representation of the relationship between standing reach and the initial velocity needed for a slam dunk.

Answer 2

The question involves applying kinematic equations to solve for the initial vertical velocity required by a basketball player to achieve a certain height during a dunk and to calculate the horizontal distance from the basket for a well-timed jump.

Step-by-step explanation:

Determining the Physics of Basketball Movements

The subject matter of this question revolves around the calculation of the vertical and horizontal components of motion in basketball.

This encompasses determining the initial velocity needed for a player to rise to a certain height when attempting a slam dunk, and calculating other aspects such as time in the air and force exerted during a jump. By using kinematic equations, we can provide step-by-step solutions to each part of the problem.

For instance, when determining the initial vertical velocity necessary for a player to rise 0.750 m above the floor, we can utilize the kinematic equation for vertical motion under constant acceleration due to gravity.

Similarly, calculating the horizontal distance from the basket requires an understanding of the player's horizontal velocity and the time taken to reach the maximum height.