A boy of 50 kg is riding a scooter of 100 kg mass at a speed of v m/s. Find v (in m/s) if the kinetic energy of the scooter and the boy is 76.8 kJ.
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Understanding the Physics Problem: Kinetic Energy of a Combined System
This problem asks us to find the speed of a system consisting of a boy and a scooter, given their individual masses and the total kinetic energy of the system. The key concept here is kinetic energy, which is the energy of motion.
When the boy is riding the scooter, they move together at the same speed. This means we can treat them as a single system with a combined mass.
Given Information
- Mass of the boy (\(m_b\)): 50 kg
- Mass of the scooter (\(m_s\)): 100 kg
- Total kinetic energy of the boy and scooter (\(KE_{total}\)): 76.8 kJ
- We need to find the speed (\(v\)) in m/s.
Calculating the Combined Mass
Since the boy and the scooter are moving together, their masses add up to form the total mass of the system.
Combined mass (\(M\)) = Mass of boy (\(m_b\)) + Mass of scooter (\(m_s\))
\[M = m_b + m_s\]
\[M = 50 \text{ kg} + 100 \text{ kg}\]
\[M = 150 \text{ kg}\]
Converting Kinetic Energy to Joules
The kinetic energy is given in kilojoules (kJ). To use it in standard physics formulas, we need to convert it to joules (J). 1 kJ = 1000 J.
\[KE_{total} = 76.8 \text{ kJ}\]
\[KE_{total} = 76.8 \times 1000 \text{ J}\]
\[KE_{total} = 76800 \text{ J}\]
Applying the Kinetic Energy Formula
The formula for kinetic energy is:
\[KE = \frac{1}{2} M v^2\]
Where:
- \(KE\) is the kinetic energy
- \(M\) is the mass of the object or system
- \(v\) is the speed of the object or system
We know the total kinetic energy (\(KE_{total}\)) and the combined mass (\(M\)). We need to solve for the speed (\(v\)).
Substitute the known values into the formula:
\[76800 \text{ J} = \frac{1}{2} \times 150 \text{ kg} \times v^2\]
\[76800 = 75 \times v^2\]
Solving for Speed (v)
Now, we rearrange the equation to find \(v^2\) and then take the square root to find \(v\).
Divide both sides by 75:
\[v^2 = \frac{76800}{75}\]
\[v^2 = 1024\]
Take the square root of both sides:
\[v = \sqrt{1024}\]
\[v = 32 \text{ m/s}\]
The speed of the scooter and the boy is 32 m/s.
Summary of Calculation Steps
| Step | Description | Calculation | Result |
|---|---|---|---|
| 1 | Calculate Combined Mass | \(50 \text{ kg} + 100 \text{ kg}\) | 150 kg |
| 2 | Convert KE to Joules | \(76.8 \times 1000 \text{ J}\) | 76800 J |
| 3 | Set up KE equation | \(76800 = \frac{1}{2} \times 150 \times v^2\) | \(76800 = 75 v^2\) |
| 4 | Solve for \(v^2\) | \(v^2 = \frac{76800}{75}\) | \(v^2 = 1024\) |
| 5 | Solve for \(v\) | \(v = \sqrt{1024}\) | 32 m/s |
The calculated speed matches option 2.
Revision Table: Key Concepts for Kinetic Energy and Mass
| Concept | Definition/Formula | Units | Importance in Problem |
|---|---|---|---|
| Kinetic Energy (KE) | Energy due to motion, \(KE = \frac{1}{2}mv^2\) | Joules (J) | Given total energy, used to find speed |
| Mass (m, M) | Amount of matter in an object | Kilograms (kg) | Boy's mass + Scooter's mass = Combined mass |
| Speed (v) | Rate of change of position | meters per second (m/s) | The unknown quantity to be calculated |
| Combined Mass | Total mass of a system of objects moving together | Kilograms (kg) | Used as 'M' in the KE formula for the system |
Additional Information: Energy and Motion
Kinetic energy is one of the fundamental forms of energy. It depends on both the mass of an object and its speed squared. This means that if you double the speed, the kinetic energy increases by a factor of four!
In this problem, we treated the boy and the scooter as a single rigid body because they move together. This is a common approach in physics problems involving systems of objects that are coupled and move with the same velocity.
Other forms of energy include potential energy (energy due to position or state, like gravitational potential energy or elastic potential energy), thermal energy (heat), chemical energy, nuclear energy, etc. The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another in a closed system.
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