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The example of the bouncing ball is used to study projectile motion in mechanics.
Projectile motion is the motion of objects that are moving near the surface of the earth in a curved path due to the effect of gravity.
At the moment of impact, the ball also experiences deformation and the coefficient of restitution, which depends on the bounciness of the ball.
The push that the ball receives from the floor at the moment of the impact causes it to bounce upwards. The moving ball gains kinetic energy when it bounces, and loses potential energy as it falls.
Newton's third law states that every force or action has an equal and opposite reaction. Hence when a force is applied to a surface, it also applies a force equal in magnitude, but in the opposite direction.
The coefficient of restitution is the ratio of the final to initial speed between two bodies after the collision. While a value of 1 indicates a perfectly elastic collision, a value of 0 indicates a perfectly inelastic collision.
In the bouncing ball example, external forces such as air resistance are assumed to be zero. Hence, the only force acting on the ball is gravity. The motion of the ball can be split into different stages depending on the direction of the velocity vector; these stages are listed below. As a general rule, when the ball is travelling in the positive direction (upwards), the velocity can be assumed to be positive. When the ball travels in the negative direction (downwards), the velocity can be assumed to be negative. The positive and negative directions must be stated in each example.
The first stage is where the ball bounces from the surface of the ground. It travels upwards towards its highest point.
The second stage is the point at which the ball decelerates, changes direction once it has reached the peak point, and starts falling to the ground.
The third stage is the point at which the ball is momentarily deformed, and bounces off the ground in an upward direction until it reaches its maximum height.
The last stage is the point at which the ball has reached its maximum displacement, decelerates, and changes the direction of motion from upwards to downwards.
These stages are continuously repeated and shown in the sequence below. it seems the ball is experiencing an oscillatory motion. In reality, the ball experiences damping, where it loses potential energy and kinetic energy as it falls. This causes the amplitude of the height to reduce over time and eventually come to a stop due to friction forces like air resistance, which are assumed to be zero in an ideal scenario. The ball is not performing a simple harmonic motion, as the acceleration is not proportional to the 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.)
At the point of maximum height, the ball momentarily has zero velocity, and the direction of velocity is changing from positive to negative. The acceleration on the ball is the acceleration of gravity, which acts downwards on the ball. At the lowest point, the ball has its minimum potential energy, and the velocity changes from negative to positive. These stages can also be represented graphically using three plots including a displacement, velocity, and acceleration vs time graph. These are illustrated below.
The first graph is a displacement vs time graph. The ball moves upwards, reaching stage 1, i.e., maximum height, and its velocity is momentarily zero. The acceleration due to gravity causes the ball to change direction and start moving downwards at stage 2.
Once the ball hits the ground, its displacement is momentarily zero. It bounces off, changing the direction of motion and again reaching its maximum height. This is also reflected in the velocity graph; the velocity is at its maximum at the minimum displacement and goes through zero at its maximum heights. The change in direction when the ball reaches the ground causes a momentary acceleration as seen in the acceleration graph (as acceleration).
A geometric sequence is a progression where each term is related to the previous term, and it is related to the previous term by a number r, which is known as the common ratio of the sequence. The last term is also known as the nth term of a geometric progression; n is the number of terms and a is the first term while Sn is the sum of the terms in the sequence as shown in the equation below.
For an infinite number of turns, another geometric sequence formula can be used. If the common ratio of the sequence is between 0 and 1, then the term r∞ would approach zero. Hence the formula for the sum of the infinite number of terms can be re-written as seen here.
Where 0 < r < 1
A real-life bouncing ball example would experience an oscillatory motion which would gradually lose energy, causing the height of the bounce to reduce over time until eventually, the ball came to a stop.
This motion can be described using a geometric sequence, as the height of the ball after each bounce depends on the initial height from which the ball fell. The last term can be the lowest height of the ball before it comes to an end as seen below. Here, the motion of a real bouncing ball is shown. Its height gradually decreases until it eventually stops moving.
A ball falls from a height of 6 metres. The ball rebounds to 38 percent of its previous height and continues to fall.
A) Find the total distance of travel until the ball hits the ground for the 5th time.
B) If this is an ideal scenario where energy is not lost and the ball continues to bounce infinitely, what is the distance of travel?
Using the geometric sequence formula, the sum of the terms which are the heights of the ball after each bound:
Finally, we need to multiply the distance found by 2, as one bounce of the ball includes both a rise and fall. Hence the final answer is:
Using the geometric sequence for an infinite sequence and substituting the given values we get:
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.
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.
Yes, as the ball is oscillating about the equilibrium position (in height) and goes back to its initial position after a period of time.
Yes, as the ball receives a force from the ground due to collision, which causes the ball to bounce off the ground.
The force that causes a ball to bounce is the reaction force described by Newton's third law of motion.
What does the motion of a bouncing ball look like?
A bouncing ball follows a projectile motion which is moving near the surface of the earth in a curved path due to the effect of gravity.
What is the coefficient of restitution?
The coefficient of restitution is the ratio of the final to the initial relative speed between two objects after they collide.
What principle describes the bouncing motion of a ball?
The push which the ball receives from the floor at the moment of impact causes it to bounce up from the surface. This is in accordance with Newton's second law.
What is the velocity of the ball at the point of maximum height?
Is a bouncing ball a simple harmonic motion? Why?
No, as the acceleration is not proportional to the displacement.
What type of motion does a bouncing ball experience?
Damping oscillatory motion
A ball falls from a height of 3 metres. The ball rebounds to 72 percent of its previous height and continues to fall. Find the total distance of travel until the ball hits the ground for the 8th time.
A ball falls from 8 metres and rebounds to 52 percent of its previous height. Find the infinite distance of travel.
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