Question

The kinetic energy of a small boat is \(15,000 \mathrm{~J}\). If the boat's speed is \(5.0 \mathrm{~m} / \mathrm{s}\), what is its mass?

Short answer

The mass of the boat is 1,200 kg.

Step-by-step solution

Understand the Kinetic Energy Formula

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

Plug Known Values into the Equation

We are given the kinetic energy \( KE = 15,000 \mathrm{~J} \) and the speed of the boat \( v = 5.0 \mathrm{~m/s} \). Substitute these values into the kinetic energy formula: \( 15,000 = \frac{1}{2} m (5.0)^2 \).

Simplify the Equation

First, calculate the square of the velocity: \( (5.0)^2 = 25 \). Then multiply by \( \frac{1}{2} \): \( \frac{1}{2} \times 25 = 12.5 \). The equation becomes \( 15,000 = 12.5m \).

Solve for Mass

Rearrange the equation to solve for mass \( m \): \( m = \frac{15,000}{12.5} \).

Calculate the Mass

Perform the division: \( m = \frac{15,000}{12.5} = 1,200 \). Therefore, the mass of the boat is \( 1,200 \mathrm{~kg} \).

Key concepts

Physics Formulas

Physics is a science that requires reliance on various formulas to understand real-world phenomenons, like motion or energy. One crucial formula is the kinetic energy equation. Kinetic energy (\( KE \)) is the energy an object possesses due to its motion. Expressed mathematically, it is calculated as:

• \( KE = \frac{1}{2} m v^2 \)

where \( m \) is the mass of the object and \( v \) is its velocity. This formula helps us determine how much energy is involved when objects are in motion, which plays a significant role in many physics problems ranging from everyday phenomena to advanced engineering applications.
Understanding and correctly applying these formulas is fundamental for students to solve problems in physics accurately. Recognizing each component and its role in the formula is key to mastering these concepts.

Problem Solving in Physics

Solving problems in physics can initially seem daunting but is manageable by following systematic steps. Here's how you can approach a physics problem:

• Comprehend the Problem: Understand what is being asked. Identify what is known (given data) and what you need to find (unknown variables).
• Select Relevant Formula: Use appropriate physics equations relevant to the problem. For kinetic energy, utilize \( KE = \frac{1}{2} m v^2 \)
• Substitute Known Values: Plug the given values into the formula. This stage often involves simple arithmetic but requires attention to units and accuracy.
• Simplify and Solve: Simplify expressions using arithmetic operations, then solve for the unknown variable.
• Reflect: Check if the result is reasonable and makes sense with the physical world.

By honing these steps, the process becomes intuitive, making physics more engaging and less intimidating.

Mass and Velocity Calculations

Mass and velocity are crucial components in calculating kinetic energy. Understanding these allows us to determine the energy an object in motion possesses.For example, consider a problem where the kinetic energy of a boat is known to be \(15,000 \, J\), and its velocity is \(5.0 \, m/s\). To find its mass, use the kinetic energy equation \( KE = \frac{1}{2} m v^2 \).

• Plug in the values: You start with \(15,000 = \frac{1}{2} m (5.0)^2\).
• Compute Velocity Squared: \((5.0)^2 = 25\).
• Simplify: Multiply by \( \frac{1}{2} \) to get \(12.5\), making the equation \(15,000 = 12.5 m\).
• Solve for mass: Rearrange to \( m = \frac{15,000}{12.5} \) which calculates to \(1,200 \, kg\).

This example illustrates the interdependence of mass and velocity in calculating kinetic energy, enhancing students' problem-solving skills in physics contexts.