Question

A superball is dropped from a \(2.00 \mathrm{~m}\) height. After it rebounds, it reaches a new height of \(1.65 \mathrm{~m}\). Assuming a constant coefficient of restitution, find the (ideal) total distance the ball will travel as it keeps bouncing.

Short answer

While mathematically the total distance that the superball would cover is infinite because of the constant coefficient of restitution, practically, factors like energy loss and air resistance will cause the ball to stop after covering a finite distance.

Step-by-step solution

Determine the coefficient of restitution

The coefficient of restitution(e) is defined as the square root of the ratio of the final height (after bounce) to the initial height (before bounce). Therefore it can be calculated as: \(e = \sqrt{\frac{H_f}{H_i}}\) where \(H_f = 1.65m\) (final height after the first bounce) and \(H_i = 2.00m\) (initial drop height). Substituting these values in, we get \(e = \sqrt{\frac{1.65}{2.00}} = 0.907\) (rounded to three decimal places).

Compute the total distance

The total distance the superball travels is the sum of an infinite geometric series where the first term (a) is the drop from the original height and the common ratio (r) is twice the coefficient of restitution (because the ball travels upward and downward during a bounce). The sum of an infinite geometric series can be calculated using the formula \(S = \frac{a}{1-r}\). In this case, \(a = H_i = 2.00m\) (initial drop height) and \(r = 2e = 2 \times 0.907 = 1.814\). Substituting these values in, we get \(S = \frac{2.00}{1 - 1.814} = -18.00m\). However, since distance cannot be negative, that is, we interpreted the series incorrectly. Instead of bouncing indefinitely, the ball bounces until its height reaches almost zero. So, the total distance is infinite.

Interpretation

In the real world, the ball won't bounce indefinitely. Each bounce depends on the previous one, and the height decreases over time. Due to factors such as air resistance and energy loss during bouncing, the height reached will eventually become zero, and the ball will stop. Therefore, while the math suggests that the ball would bounce infinitely, in practice, this won't happen. The total distance travelled is thus considered to be infinite because of the assumption of a constant coefficient of restitution. But in reality, the total distance will be finite and less than the mathematically determined value.

Key concepts

Infinite Geometric Series

When a superball bounces, each successive height it reaches can be considered terms in an infinite geometric series, assuming a perfect environment with no energy loss. An infinite geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a constant number, known as the ratio.

Understanding infinite geometric series is crucial when predicting the total distance traveled by an object like the superball in our exercise. Using the formula for the infinite geometric series sum, \( S = \frac{a}{1-r} \), where \( a \) is the first term and \( r \) is the common ratio, we can calculate the ideal total distance, despite it being infinite. Additionally, the condition for convergence, \( |r| < 1 \), must be satisfied for the series to have a finite sum, which is not the case here due to the ratio being greater than one.

Kinetic Energy Conservation

Kinetic energy conservation plays a vital role in understanding elastic collisions. In an ideal scenario, where there is no energy loss, the kinetic energy of a system is conserved before and after a collision.

In the context of our superball problem, considering kinetic energy is conserved in an elastic collision helps to determine the coefficient of restitution. This coefficient relates the relative speeds of two objects in a collision, and hinting that the potential energy lost when the ball hits the ground is converted back to kinetic energy, propelling it back into the air. Although this simplification ignores real-world factors like air resistance and deformation energy, it provides a valuable theoretical framework for understanding the ball's motion.

Elastic Collision

An elastic collision is one in which both momentum and kinetic energy are conserved. In the idealized exercise, the bounce of the superball is considered to be perfectly elastic, meaning no energy is lost in the form of sound, heat, or deformation during the collision with the ground.

However, in real life, some energy is always transformed into non-recoverable forms. To accommodate for the imperfectly elastic nature of real-world collisions, the concept of the coefficient of restitution is introduced to measure the elasticity of the collision. This coefficient is a ratio of the relative velocities before and after the collision and, in perfect elastic collisions, equals one. With the value calculated from the given heights, we can assess the efficiency of the energy conversion during the superball's bounces, bearing in mind that in practical applications, the ball will eventually cease to bounce due to energy dissipation.