# Bouncing Ball Example

The bouncing ball is a classic example used to study projectile motion in mechanics. Projectile motion refers to the curved path objects near the earth's surface follow due to gravity. When the ball hits the ground, it deforms and bounces back up because of the force applied by the floor. As the ball bounces, it gains kinetic energy and loses potential energy. Newton's third law states that every action has an equal and opposite reaction, and the coefficient of restitution measures the ratio of final to initial speed after a collision. The motion of the ball can be divided into stages, each with its own direction and velocity. As the ball bounces, it experiences damping, which reduces its amplitude and eventually brings it to a stop due to friction forces like air resistance. The ball's motion is not simple harmonic because its acceleration is not proportional to its displacement from an equilibrium position.

(The upward direction was assumed to be positive in this example. This can either be assumed and chosen, or it can be stated in a question.)

When the ball reaches its maximum height, it briefly stops moving and changes the direction of its velocity from positive to negative. At the lowest point, the ball has its minimum potential energy and changes its velocity from negative to positive. The ball's acceleration is always downwards due to gravity. These stages can be shown visually using three graphs: displacement, velocity, and acceleration vs. time. See the illustrations below.

The displacement of the ball at 50 seconds can be found by using the area under the graph, which is equal to the displacement. The area of the triangle can be found using the formula: Area = (1/2) × base × height. The velocity of the ball before it hits the ground from a height of three metres can be found by using the conservation of energy. We equate the potential energy and the kinetic energy, and rearrange with respect to velocity: v = √2gh, where g is the acceleration due to gravity and h is the height of the ball.

## What is a geometric sequence?

A geometric sequence is a sequence where each term is related the previous term by a common ratio, denoted by r. The nth term of a geometric sequence is given by an = ar^(n-1), where a is the first term of the sequence. The sum of n terms of a geometric sequence is given by the formula Sn = a(1-r^n)/(1-r). For an infinite geometric sequence with a common ratio between 0 and 1, the sum of an infinite number of terms can be calculated using the formula S∞ = a/(1-r). This is because as n approaches infinity, r^n approaches zero, making the expression (1-r^n)/(1-r) approach 1/(1-r).

### Relation of geometric sequence and a bouncing ball

To model the height of a bouncing ball using a geometric sequence, we can assume that the ball bounces to a fraction of its previous height with each bounce. Let the initial height of the ball be h, and let the ball bounce to a fraction of k of its previous height with each bounce. Then, the height of the ball after n bounces is given by hn = h(k)^n. The sequence hn is a geometric sequence with first term h and common ratio k. The height of the ball after the nth bounce is proportional to the nth power of the common ratio k. As the ball bounces, the height of the ball decreases over time until it eventually comes to a stop. This loss of energy is due to various factors such as air resistance and the ball's elasticity. Therefore, the sequence hn approaches zero as n approaches infinity.

## Height-Time graph of a bouncing ball

For Solution A, we are given that the ball falls from a height of 6 metres and rebounds to 38% of its previous height. This means that the common ratio k = 0.38. We need to find the total distance traveled by the ball until it hits the ground for the 5th time. To do this, we use the formula for the sum of the first n terms of a geometric sequence:

Sn = a(1-k^n)/(1-k)

Here, a is the first term of the sequence, which is 6 metres, and n is 5. Substituting these values and solving for Sn, we get:

Sn = 6(1-0.38^5)/(1-0.38) = 8.456 metres

However, this distance includes both the upward and downward travel of the ball. To find the total distance traveled, we need to multiply this value by 2, giving us:

Total distance = 2 x 8.456 = 16.912 metres

For Solution B, we are asked to find the distance of travel if the ball bounces infinitely, without losing any energy. In this case, we can use the formula for the sum of an infinite geometric sequence:

S∞ = a/(1-k)

Substituting the given values, we get:

S∞ = 6/(1-0.38) = 9.677 metres

Again, this distance includes both the upward and downward travel of the ball, so the total distance traveled would be:

Total distance = 2 x 9.677 = 19.354 metres

Therefore, the total distance traveled by the ball in this ideal scenario would be 19.354 metres.

## Bouncing Ball Example - Key takeaways

The motion of a bouncing ball can be analyzed using various tools and in mechanics such as projectile motion, Newton's laws of motion, energy conservation, and kinematics. Bouncing ball motion can be represented using displacement-time, velocity-time, and acceleration-time graphs. The displacement-time graph would form a series of parabolic curves representing the motion of the ball in each bounce. The velocity-time graph would show the velocity of the ball as it moves up and down, with the velocity becoming zero at the highest point of each bounce. The acceleration-time graph would show that the acceleration of the ball is constant during its fall and rise, but changes direction at the highest point of each bounce. Using these graphs, the motion of the ball can be analyzed and predicted.

## Bouncing Ball Example

Is a bouncing ball an example of simple harmonic motion?

No, the bouncing ball example is not an example of simple harmonic motion. Its high order and functions achieved with differential and integral operations can't fit any circle, because circles must cover constant speed in simple harmonic motion.

Is a bouncing ball an example of potential energy?

Yes, as elastic potential energy causes the ball to bounce off the ground and is converted into kinetic energy once the ball is in the air, causing it to move.

Is a bouncing ball an example of oscillatory motion?

Yes, as the ball is oscillating about the equilibrium position (in height) and goes back to its initial position after a period of time.

Is a bouncing ball an example of Newton's third law?

Yes, as the ball receives a force from the ground due to collision, which causes the ball to bounce off the ground.

What is the force that causes a ball to bounce?

The force that causes a ball to bounce is the reaction force described by Newton's third law of motion.