Question

Rank Four joggers have the following masses and speeds: \begin{tabular}{|c|c|c|} \hline Jogger & Mass & Speed \\ \hline A & \(m\) & \(v\) \\ \hline B & \(m / 2\) & \(3 v\) \\ \hline C & \(3 m\) & \(v / 2\) \\ \hline D & \(4 m\) & \(v / 2\) \\ \hline \end{tabular} Rank the joggers in order of increasing kinetic energy. Indicate ties where appropriate.

Short answer

The order is C, A, D, B.

Step-by-step solution

Understanding Kinetic Energy Formula

The formula for kinetic energy (KE) is given by \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the speed of the object.

Calculate Kinetic Energy for Jogger A

Jogger A has mass \( m \) and speed \( v \). Thus, the kinetic energy for A is \( KE_A = \frac{1}{2}m v^2 \).

Calculate Kinetic Energy for Jogger B

Jogger B has mass \( \frac{m}{2} \) and speed \( 3v \). Substitute these values into the kinetic energy formula: \( KE_B = \frac{1}{2} \left( \frac{m}{2} \right) (3v)^2 = \frac{1}{2} \times \frac{m}{2} \times 9v^2 = \frac{9}{4}mv^2 \).

Calculate Kinetic Energy for Jogger C

Jogger C has mass \( 3m \) and speed \( \frac{v}{2} \). Substitute these values into the kinetic energy formula: \( KE_C = \frac{1}{2} \times 3m \times \left( \frac{v}{2} \right)^2 = \frac{1}{2} \times 3m \times \frac{v^2}{4} = \frac{3}{8}mv^2 \).

Calculate Kinetic Energy for Jogger D

Jogger D has mass \( 4m \) and speed \( \frac{v}{2} \). Substitute these values into the kinetic energy formula: \( KE_D = \frac{1}{2} \times 4m \times \left( \frac{v}{2} \right)^2 = \frac{1}{2} \times 4m \times \frac{v^2}{4} = mv^2 \).

Compare Kinetic Energies

Determine the relative magnitudes of kinetic energies: \( KE_C = \frac{3}{8}mv^2 \), \( KE_A = \frac{1}{2}mv^2 \), \( KE_B = \frac{9}{4}mv^2 \), and \( KE_D = mv^2 \). Therefore, the ranking in increasing order is C, A, D, B.

Key concepts

Joggers

In our exercise, we have four joggers labeled A, B, C, and D. Each jogger has different combinations of mass and speed. Understanding these parameters is key to solving the problem. Jogger A has a baseline mass \( m \) and speed \( v \). Jogger B's mass is half of A but moves three times faster. Jogger C is three times heavier than A but jogs at half the speed. Jogger D carries four times the weight of A and jogs at half the speed. Each combination affects the energy communicated by their movement.

Masses and Speeds

Mass and speed are fundamental in determining how much kinetic energy each jogger has.

• Mass is the measure of how much matter is in a jogger, influencing how much energy they keep when moving.
• Speed determines how fast the jogger is moving. In the kinetic energy formula, speed plays a crucial role as it gets squared.

For Jogger A, the mass \( m \) is simply 'as is,' and the speed is \( v \). For Jogger B, with mass \( \frac{m}{2} \) and greater speed \( 3v \), these elements culminate into more kinetic energy. Jogger C has a greater mass \( 3m \), reducing speed to \( \frac{v}{2} \), while Jogger D is the heaviest at \( 4m \) but jogs slowly at \( \frac{v}{2} \). Each jogger's mass and speed will decisively impact their position in the kinetic energy rank.

Kinetic Energy Formula

The kinetic energy formula is a handy tool for calculating the energy of moving objects, such as our joggers. It is given by:\[KE = \frac{1}{2} mv^2\]In this formula:

• \( KE \) represents kinetic energy, measured in joules.
• \( m \) represents mass, measured in kilograms.
• \( v \) represents speed, measured in meters per second.

A critical point to note is that speed is squared in this equation, meaning that even a small increase in speed results in a larger increase in kinetic energy than an equivalent increase in mass.

Problem-Solving Steps

Solving this problem involves a few essential steps:

• Step 1: Identify each jogger's properties – mass and speed.
• Step 2: Use the kinetic energy formula for each jogger. Substituting their specific mass and speed, as shown:
  • Jogger A: \( KE_A = \frac{1}{2}m v^2 \)
  • Jogger B: \( KE_B = \frac{9}{4}mv^2 \)
  • Jogger C: \( KE_C = \frac{3}{8}mv^2 \)
  • Jogger D: \( KE_D = mv^2 \)
• Step 3: Rank these calculated values from smallest to largest: \( KE_C < KE_A < KE_D < KE_B \).

By following these structured steps, determining each jogger's kinetic energy becomes systematic and straightforward.