Question

Calculate 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 Kinetic Energy Formula

The kinetic energy (KE) of an object is given by the formula \( KE = \frac{1}{2} m v^2 \), where \( m \) is the mass of the object and \( v \) is its velocity or speed. We need to solve this formula for \( v \).

Rearrange the Formula to Solve for Velocity

Starting with the kinetic energy formula \( KE = \frac{1}{2} m v^2 \), we solve for speed \( v \). Rearranging gives \( v^2 = \frac{2 \times KE}{m} \).

Substitute Known Values

Substitute the given values into the rearranged formula. Here \( KE = 7.8 \text{ J} \) and \( m = 0.27 \text{ kg} \). Thus, \( v^2 = \frac{2 \times 7.8}{0.27} \).

Calculate the Square of the Velocity

Calculate \( v^2 = \frac{15.6}{0.27} = 57.78 \).

Calculate the Velocity

Find the velocity by taking the square root of \( 57.78 \). Hence, \( v = \sqrt{57.78} \approx 7.6 \text{ m/s} \).

Key concepts

Speed Calculation

When we talk about speed calculation in physics problems, it often involves determining how quickly an object moves from one point to another. Speed is a scalar quantity, meaning it does not have a direction—it simply measures how fast something is moving. To calculate speed, you generally use the formula:

• The basic formula for speed is:\[\text{Speed} = \frac{\text{Distance}}{\text{Time}}\]

However, in scenarios involving kinetic energy, such as in our volleyball example, we need to use a different approach.Here, the speed is derived from kinetic energy, where the formula:\[v^2 = \frac{2 \, KE}{m}\]allows us to calculate the square of the speed. To find speed \( v \), take the square root of both sides of this equation.

Physics Problems

Physics problems often involve applying key concepts of energy, force, and motion to find unknown quantities. In the exercise about the volleyball, we are exploring the concept of kinetic energy, which is the energy an object holds due to its motion. When approaching physics problems, follow these tips for better understanding:

• Carefully read the problem to understand what is given and what needs to be found. Identify known values and the quantities to solve.
• Apply relevant physics formulas that connect the given values with the unknowns.
• Check units of measurement to ensure your calculations use consistent units throughout.

Such systematic approaches help break down complex problems into manageable steps, as seen in our volleyball speed calculation.

Energy Formulas

Energy formulas are key tools in physics that allow scientists and students alike to quantify how energy moves and changes form. In kinetic energy problems, we use specific formulas to express how mass and speed relate to the energy of motion.Kinetic energy, represented as:\[KE = \frac{1}{2} m v^2\]where \( KE \) is kinetic energy, \( m \) is mass, and \( v \) is velocity. This formula shows that kinetic energy increases with both mass and speed. High speed and large mass lead to higher kinetic energy.In our example, rearranging the formula helped us find the speed of the volleyball when its kinetic energy and mass are known. Such rearrangements highlight the versatile nature of physics formulas in solving diverse problems.