Question

A ball is thrown up in the air, reaching a height of \(5.00 \mathrm{~m}\). Using energy conservation considerations, determine its initial speed.

Short answer

Answer: The initial speed of the ball is approximately 9.95 m/s.

Step-by-step solution

Initial Mechanical Energy

Initially, the ball possesses only kinetic energy and no potential energy, as it has yet to gain any height. So the initial mechanical energy (E1) will be equal to the initial kinetic energy (KE1). The kinetic energy is given by the formula: KE = 0.5 * m * v^2, where m is the ball's mass and v is its initial speed. So, E1 = KE1 = 0.5 * m * v^2

Mechanical Energy at the Highest Point

At the ball's highest point, it momentarily comes to rest, losing all its kinetic energy. However, it gains gravitational potential energy (PE), which is given by the formula: PE = m * g * h, where m is the ball's mass, g is the gravitational acceleration (9.81 m/s²), and h is the height. So, the mechanical energy at the highest point (E2) will be equal to the gravitational potential energy (PE). E2 = PE = m * g * h

Conservation of Mechanical Energy

According to the conservation of mechanical energy principle, the initial mechanical energy (E1) should equal the mechanical energy at the highest point (E2). So, 0.5 * m * v² = m * g * h

Solving for the Initial Speed

Now, we can solve for the initial speed v: 0.5 * m * v² = m * g * h Divide both sides by m: 0.5 * v² = g * h Now, we can plug in the values for g (gravitational acceleration) and h (height): 0.5 * v² = 9.81 * 5.00 Next, solve for v²: v² = 9.81 * 5.00 * 2 Now, take the square root of both sides: v = sqrt(9.81 * 5.00 * 2) Calculating the value: v ≈ 9.95 m/s

Final Answer

The initial speed of the ball is approximately 9.95 m/s.

Key concepts

Understanding Kinetic Energy

When an object is in motion, it inherently possesses a type of energy known as kinetic energy. It's the energy related to the object's velocity and mass. Specifically, the kinetic energy (KE) of an object can be quantified by the equation:
\[ KE = \frac{1}{2} m v^2 \] where \(m\) stands for the mass of the object and \(v\) represents the velocity or speed at which the object is moving. Importantly, kinetic energy is always a positive value, as it is proportional to the square of the velocity. This energy can be transformed into other types of energy under different circumstances, which plays a crucial role in the principle of conservation of mechanical energy.
One typical real-life example is a ball thrown into the air. At the moment it leaves your hand, the ball has the maximum kinetic energy, depending solely on its mass and initial speed (before it starts slowing down due to gravity).

Gravitational Potential Energy Explained

Gravitational potential energy (PE) is the energy an object possesses because of its position in a gravitational field. It is dependent on the object's mass, the height above a reference point, and the acceleration due to gravity. The formula for calculating gravitational potential energy is:
\[ PE = mgh \] Here, \(m\) is the object's mass, \(g\) is the acceleration due to gravity (on Earth this is approximately \(9.81 \, \text{m/s}^2\)), and \(h\) is the height of the object from the reference point.
Consider a ball tossed upwards; upon reaching its highest point, all the kinetic energy it had has been converted into gravitational potential energy. This is because at the peak, its speed is momentarily zero, and so is its kinetic energy. This exact point, where the ball stops rising, represents its maximum potential energy.

Calculating Initial Speed

To figure out the initial speed of an object, it often involves reworking the energy relationships. For projectile motion, like a ball thrown into the air, the task is to find the speed at which it was initially thrown (the initial velocity). Assuming no air resistance and that energy is conserved, we can use the mechanical energy conservation equation related to kinetic and potential energy here.
The steps involve setting the initial kinetic energy equal to the potential energy at the highest point, then solving for the velocity. This calculation is based on the principle that the sum of kinetic and potential energy at the beginning is equal to the sum at any other point, assuming no energy is lost. In the case of the ball that reaches a height of \(5.00 \, \text{m}\), we follow these steps using the formulas for kinetic and gravitational potential energy to find the initial speed. This is an essential calculation in mechanics, often used to predict the motion of objects under the force of gravity.

The Principle of Energy Conservation

The principle of conservation of mechanical energy states that in a closed system with no external forces, like friction or air resistance, the total mechanical energy remains constant. Mechanical energy is the sum of kinetic and potential energy in a system. So, as an object moves within this closed system, its energy may convert from one form to another, but the total amount of energy remains unchanged.
In our ball example, as the ball rises, its kinetic energy is transformed into potential energy. The initial mechanical energy when the ball was thrown (purely kinetic) equals the mechanical energy at its highest point (purely potential), illustrating this principle. It's a powerful concept that allows us to solve for unknowns such as initial speed, maximum height, and velocity at different points in an object's trajectory. Understanding this principle is fundamental for studying physics and various engineering applications.