91Ó°ÊÓ

When you hear the term free fall, it refers to any motion where only the force of gravity is acting on an object. Imagine dropping a ball from a high place, just like in our exercise. This ball is in free fall as it travels toward the ground, accelerating due to gravity without any other forces (like air resistance) slowing it down.
Gravity is a constant force on Earth, pulling objects down at an acceleration of 9.81 meters per second squared (m/s²). When the sphere in our exercise is dropped from a 2-meter height, it gains speed as it falls. Before it hits the ground, its velocity can be calculated using the formula for free fall velocity: \[ v = \sqrt{2gy} \]where

• \( v \) is the velocity,
• \( g \) is the gravitational acceleration (9.81 m/s²),
• \( y \) is the height (drop height).

This means that the velocity of the sphere just before it hits the spring floor will be approximately 6.26 m/s. Understanding free fall helps us predict the speed at which the sphere impacts the surface.

The coefficient of restitution (\(e\)) is a measure of how much energy of motion (or kinetic energy) remains after a collision. It is the ratio of the velocities after and before the impact. Basically, it tells us how "bouncy" two colliding bodies are.
A coefficient of restitution equal to 1 means a perfectly elastic collision—no kinetic energy is lost. Conversely, a coefficient of zero indicates a perfectly inelastic collision, where objects stick together and most kinetic energy is lost.
In our problem, the sphere and the spring floor have a coefficient of restitution of 0.6, indicating that some energy is lost in the collision. After the sphere hits the surface and rebounds, the new velocity is reduced because of this energy loss. The formula used to calculate the rebound velocity is: \[v' = e \times v\]Where:

• \(v'\) is the velocity after the rebound,
• \(e\) is the coefficient of restitution (0.6),
• \(v\) is the initial velocity before impact (6.26 m/s).

This calculation aids in determining how high the sphere will bounce back.

Springs have a certain stiffness, described by the spring constant \(k\). This constant represents how much force a spring applies per unit of extension or compression. A high \(k\) indicates a stiffer spring.
In this scenario, the spring floor has a stiffness of 120 kN/m. This robust spring system absorbs the fall energy and returns it during rebound, much like a trampoline.
To find the maximum force the springs exert, we need to consider the reduction of energy in the spring system. The potential energy stored in a compressed spring is given by:\[ \frac{1}{2}kx^2 \]where:

• \(k\) is the spring constant (120,000 N/m),
• \(x\) is the spring compression.

In our example, the spring compresses slightly to absorb the impact. By solving the energy conservation equation, we find the spring compression necessary to stop the sphere and then calculate the maximum spring force using:\[ F = kx \]This force is a key value in understanding spring dynamics in mechanics.

The principle of energy conservation is one of the core ideas in physics. It states that energy cannot be created or destroyed, only transferred or transformed. In the sphere and spring system, the principle helps us track energy changes through different forms.
Initially, the sphere has gravitational potential energy when held at a height of 2 meters. As it falls and its altitude decreases, this potential energy converts to kinetic energy, increasing the sphere's speed.
Upon colliding with the spring floor, the kinetic energy helps compress the springs, becoming stored as elastic potential energy. When the sphere rebounds, some of this stored energy returns to the sphere, propelling it upward. Throughout this interchange:

• Potential energy becomes kinetic as the sphere falls,
• Kinetic energy transfers to elastic potential energy as the spring compresses,
• Elastic potential energy partly converts back to kinetic energy as the sphere bounces up.

Understanding this flow and conversion of energy helps us quantify the maximum height reached post-rebound and the force exerted by the spring system.