Question
Find the kinetic energy of a 4 kg particle whose velocity is \( (\hat{i}-\hat{j}) \mathrm{ms} \)
Ask by Knight Bright. in Ghana
Jan 18,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The kinetic energy of the particle is 4 Joules.
Solution
To calculate the **kinetic energy** of a particle, you can use the formula:
\[
K = \frac{1}{2} m v^2
\]
where:
- \( K \) is the kinetic energy,
- \( m \) is the mass of the particle,
- \( v \) is the speed of the particle.
Given:
- Mass (\( m \)) = **4 kg**
- Velocity vector (\( \vec{v} \)) = \( \hat{i} - \hat{j} \, \text{m/s} \)
**Step 1: Determine the Speed (\( v \))**
The speed is the magnitude of the velocity vector.
\[
v = |\vec{v}| = \sqrt{(1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \, \text{m/s}
\]
**Step 2: Calculate the Kinetic Energy (\( K \))**
\[
K = \frac{1}{2} \times 4 \, \text{kg} \times (\sqrt{2} \, \text{m/s})^2
\]
\[
K = \frac{1}{2} \times 4 \times 2
\]
\[
K = \frac{1}{2} \times 8 = 4 \, \text{Joules}
\]
**Final Answer:**
The kinetic energy of the particle is **4 Joules**.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To find the kinetic energy of the particle, we use the formula: \[ KE = \frac{1}{2} mv^2 \] where \( m \) is the mass of the particle and \( v \) is its speed. Here, the mass \( m = 4 \) kg, and the velocity vector \( \vec{v} = \hat{i} - \hat{j} \). First, we calculate the speed \( v \): \[ v = \sqrt{(1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \, \text{m/s} \] Now, substituting the values into the kinetic energy formula: \[ KE = \frac{1}{2} \cdot 4 \cdot (\sqrt{2})^2 = \frac{1}{2} \cdot 4 \cdot 2 = 4 \, \text{J} \] So, the kinetic energy of the particle is 4 Joules.
