Question

A 1.50 -kg ball has a speed of \(20.0 \mathrm{~m} / \mathrm{s}\) when it is \(15.0 \mathrm{~m}\) above the ground. What is the total energy of the ball?

Short answer

Answer: The total energy of the ball is 520.725 J.

Step-by-step solution

Calculate the potential energy (PE)

We can calculate the potential energy using the formula: \(\text{PE} = m \cdot g \cdot h\), where \(m\) is the mass, \(g\) is the gravitational acceleration (approximately \(9.81 \mathrm{~m/s^2}\)), and \(h\) is the height above the ground. In our case, \(m = 1.50 \mathrm{~kg}\), \(g = 9.81 \mathrm{~m/s^2}\), and \(h = 15.0 \mathrm{~m}\). Therefore, \(\text{PE} = 1.50 \cdot 9.81 \cdot 15.0 = 220.725 \mathrm{~J}\) (Joules).

Calculate the kinetic energy (KE)

We can calculate the kinetic energy using the formula: \(\text{KE} = \frac{1}{2} \cdot m \cdot v^2\), where \(m\) is the mass and \(v\) is the speed. In our case, \(m = 1.50 \mathrm{~kg}\) and \(v = 20.0 \mathrm{~m/s}\). Therefore, \(\text{KE} = \frac{1}{2} \cdot 1.50 \cdot (20.0)^2 = 300 \mathrm{~J}\) (Joules).

Calculate the total energy

The total energy is the sum of potential energy and kinetic energy: \(\text{Total Energy} = \text{PE} + \text{KE}\). We calculated \(\text{PE} = 220.725 \mathrm{~J}\) and \(\text{KE} = 300 \mathrm{~J}\) in the previous steps. Therefore, \(\text{Total Energy} = 220.725 + 300 = 520.725 \mathrm{~J}\) (Joules). The total energy of the ball is \(520.725 \mathrm{~J}\).

Key concepts

Potential Energy

Potential energy is essentially stored energy, waiting to be converted into another form. It is dependent on the position or state of an object. For a ball that is lifted to a certain height, we calculate its gravitational potential energy.

In the exercise, the potential energy (PE) of a 1.50-kg ball lifted 15.0 m above the ground can be calculated using the formula:

\[ PE = m \times g \times h \]

Here, \(m\) represents the mass of the ball, \(g\) denotes the acceleration due to gravity (\(9.81 \, \text{m/s}^2\)), and \(h\) signifies the height. Multiplying the ball's mass by gravity's pull and the height gives the potential energy, which is the energy the ball has due to its position above the Earth. The computed result is \(220.725 \, \text{J}\) (joules), which considers how much work would be needed to lift the ball to that height or the amount of energy that would be released if the ball fell from that height.

Kinetic Energy

Kinetic energy, on the other hand, represents the energy of motion. Any object that is moving possesses kinetic energy, which increases with the object's speed and mass. The kinetic energy (KE) of an object is calculated with the formula: \[ KE = \frac{1}{2} \times m \times v^2 \]

In the context of the given exercise, where a 1.50-kg ball is moving at a speed of 20.0 m/s, its kinetic energy can be worked out using its mass \(m\) and the square of its velocity \(v\). The calculation results in a kinetic energy of \(300 \, \text{J}\), reflecting how much energy is present due to the ball's movement. This energy could do work on other objects it interacts with, for instance, if the ball hits a surface and causes a deformation.

Mechanical Energy

Mechanical energy is the sum of potential and kinetic energies, representing the total energy of motion and position of an object. It's a useful concept in physics because it helps us understand how objects move and interact within a system.

To find the total mechanical energy in the exercise, we simply add the potential energy (\(PE\)) to the kinetic energy (\(KE\)), as shown in this equation: \[ \text{Total Energy} = PE + KE \]

With the ball's potential energy computed at \(220.725 \, \text{J}\) and kinetic energy at \(300 \, \text{J}\), the total mechanical energy is \(520.725 \, \text{J}\). This total reflects the energy available to the ball due to both its elevation and its motion. The beauty of the conservation of energy principle is that this total mechanical energy of the system remains constant if no external forces perform work on or remove energy from the system.