Question

A \(0.27-\mathrm{kg}\) volleyball has a kinetic energy of \(7.8 \mathrm{~J}\). What is the speed of the volleyball?

Short answer

The speed of the volleyball is approximately 7.6 m/s.

Step-by-step solution

Understanding the problem

The problem provides the mass of the volleyball, which is 0.27 kg, and its kinetic energy, which is 7.8 J. We are asked to find the speed of the volleyball.

Writing the kinetic energy formula

The formula for kinetic energy is given by:\[KE = \frac{1}{2} m v^2\]where \(KE\) is the kinetic energy, \(m\) is the mass, and \(v\) is the velocity.

Solving the kinetic energy formula for velocity

We need to solve for \(v\) in the kinetic energy equation. Rearrange the formula to solve for \(v\):\[v = \sqrt{\frac{2 \, KE}{m}}\]

Substituting the known values

Substitute the known values of \(KE = 7.8 \, J\) and \(m = 0.27 \, kg\) into the formula:\[v = \sqrt{\frac{2 \times 7.8}{0.27}}\]

Calculating the speed

Calculate the value inside the square root first:\[v = \sqrt{\frac{15.6}{0.27}} = \sqrt{57.7777...}\]

Final computation

Compute the square root to find the speed:\[v \approx 7.6 \, m/s\]

Key concepts

Velocity

Velocity is a fundamental concept in physics. It describes the speed at which an object moves in a specific direction. Unlike speed, which only tells us how fast something is moving, velocity includes both the object’s speed and the direction of its travel.
Imagine being in a car traveling north at 50 km/h. Your speed is 50 km/h, but your velocity is 50 km/h north. To find velocity in physics calculations, you often need the object's kinetic energy and mass.
For instance, in our volleyball problem, we derived the volleyball's speed from its kinetic energy using the kinetic energy formula. This ties velocity into a broader context, helping us understand more about the object's motion as it relates to its energy.

Mass

Mass is a measure of the amount of matter in an object. It is one of the key properties that determine how an object will respond to forces. In our volleyball exercise, the mass plays a critical role in calculating the velocity from kinetic energy.
Mass is constant for an object and doesn't change regardless of where it is in the universe (unlike weight, which can change with gravity alterations). It appears in mechanics and energy calculations, often symbolized by the letter 'm'.
In the kinetic energy formula \[ KE = \frac{1}{2} m v^2 \] the mass of the volleyball (0.27 kg), is crucial as it scales the effect of the velocity on the kinetic energy. This shows how mass directly influences the energy and motion of objects.

Physics Formulas

Physics formulas provide a structured way to calculate various physical properties such as velocity and energy. They are essential in understanding how different factors like force, mass, velocity, and energy interrelate.
In our problem, we used the kinetic energy formula \[ KE = \frac{1}{2} m v^2 \] to find the volleyball’s speed. By rearranging this formula, we were able to isolate and solve for velocity: \[ v = \sqrt{\frac{2 \, KE}{m}} \]. This showcases the utility of physics formulas in solving real-world problems.
These formulas aren't just numbers and letters; they reveal the relationships and principles governing physical phenomena, allowing us to predict and understand the behavior of objects under different conditions.