In computer graphics you are tasked with creating a keyframe animation of a bouncing ball. The animation should

start with the ball at a rest position $($Frame $1),$ then it should rise to a maximum height $($Frame

$15),$ bounce back to a lower height $($Frame $30),$ and finally come to a rest again $($Frame $45).$

The total duration of the animation is $3$ seconds, and the frame rate is $15$ frames per second

$($fps$).$

a$)$ Calculate the positions of the ball at specific intermediate frames between the

keyframes. For instance, determine the position of the ball at Frame $5,$ Frame $10,$

Frame $20,$ Frame $25,$ and so on$,$ up to Frame $45.$ Consider the physics of motion,

including acceleration due to gravity.

b$)$ Implement easing functions to make the motion look more natural. Describe how

you would apply ease$-$in and ease$-$out functions to the ball's animation, both during

its upward and downward motion. Explain the purpose of using these functions.

c$)$ Create a graph that visually represents the position of the ball over time. The x$-$axis

should represent time $($in seconds$),$ and the y$-$axis should represent the vertical

position of the ball. Plot the positions at keyframes and intermediate frames. Label

the graph with frame numbers and the ball's height in units.