If an object of mass 10 kg is moving with a uniform speed of 10 m/s, then the linear momentum and the kinetic energy of the object, respectively, are
100 N.s and 500 J
Calculating Linear Momentum and Kinetic Energy
This problem asks us to find the linear momentum and kinetic energy of an object given its mass and speed. We need to apply the standard formulas for these two physical quantities.
Understanding the Concepts
- Linear Momentum (p): This is a measure of the mass in motion. It is defined as the product of an object's mass and its velocity. The formula is \(p = m \times v\), where \(m\) is mass and \(v\) is velocity. The unit is kg⋅m/s, which is equivalent to Newton-second (N⋅s).
- Kinetic Energy (KE): This is the energy possessed by an object due to its motion. It is defined as half of the product of an object's mass and the square of its speed. The formula is \(KE = \frac{1}{2} m v^2\), where \(m\) is mass and \(v\) is speed. The unit is Joules (J).
Given Information
From the question, we are given:
- Mass of the object, \(m = 10 \, \text{kg}\)
- Uniform speed of the object, \(v = 10 \, \text{m/s}\)
Since the object is moving with a uniform speed in a specific direction, we can consider the magnitude of the velocity to be equal to the speed for calculating momentum.
Step-by-Step Calculation
Step 1: Calculate Linear Momentum
We use the formula for linear momentum:
\(p = m \times v\)
Substitute the given values:
\(p = 10 \, \text{kg} \times 10 \, \text{m/s}\)
Calculate the product:
\(p = 100 \, \text{kg} \cdot \text{m/s}\)
The unit kg⋅m/s is equivalent to N⋅s. So, the linear momentum is \(100 \, \text{N} \cdot \text{s}\).
Step 2: Calculate Kinetic Energy
We use the formula for kinetic energy:
\(KE = \frac{1}{2} m v^2\)
Substitute the given values:
\(KE = \frac{1}{2} \times 10 \, \text{kg} \times (10 \, \text{m/s})^2\)
First, calculate the square of the speed:
\(v^2 = (10 \, \text{m/s})^2 = 100 \, \text{m}^2/\text{s}^2\)
Now substitute this back into the KE formula:
\(KE = \frac{1}{2} \times 10 \, \text{kg} \times 100 \, \text{m}^2/\text{s}^2\)
Calculate the product:
\(KE = 5 \, \text{kg} \times 100 \, \text{m}^2/\text{s}^2\)
\(KE = 500 \, \text{kg} \cdot \text{m}^2/\text{s}^2\)
The unit kg⋅m²/s² is equivalent to Joules (J). So, the kinetic energy is \(500 \, \text{J}\).
Summary of Results
- Linear Momentum = \(100 \, \text{N} \cdot \text{s}\)
- Kinetic Energy = \(500 \, \text{J}\)
The question asks for the linear momentum and kinetic energy respectively. Therefore, the answer should be presented as \(100 \, \text{N} \cdot \text{s}\) and \(500 \, \text{J}\).
Matching with Options
Let's compare our calculated values with the given options:
| Option | Linear Momentum | Kinetic Energy | Matches Calculation? |
|---|---|---|---|
| 1 | 100 N.s | 500 J | Yes |
| 2 | 100 N.s | 1000 J | No (KE is wrong) |
| 3 | 200 N.s | 500 J | No (Momentum is wrong) |
| 4 | 200 N.s | 1000 J | No (Both are wrong) |
Option 1 correctly matches our calculated values for both linear momentum and kinetic energy.
Revision Table: Key Physics Concepts
| Concept | Formula | Units |
|---|---|---|
| Linear Momentum (\(p\)) | \(p = m v\) | kg⋅m/s or N⋅s |
| Kinetic Energy (KE) | \(KE = \frac{1}{2} m v^2\) | kg⋅m²/s² or J (Joules) |
| Mass (\(m\)) | - | kg (kilograms) |
| Velocity (\(v\)) / Speed (\(v\)) | - | m/s (meters per second) |
Additional Information: Momentum and Energy
Linear momentum is a vector quantity, meaning it has both magnitude and direction. In this problem, we were given speed, which is the magnitude of velocity. For kinetic energy, which is a scalar quantity, we only need the magnitude of velocity (speed).
The concept of momentum is related to Newton's laws of motion, particularly the second law (\(F_{net} = \frac{\Delta p}{\Delta t}\)). Energy, including kinetic energy, is a scalar quantity that represents the capacity to do work. The work-energy theorem relates the net work done on an object to its change in kinetic energy.
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