Write the momentum vector of a 10 kg ball moving with velocity 40 m/s SE (-45 deg) in standard unit vector notation. Also, calculate the x- and y-components of the momentum.
Write the momentum vector of a 10 kg ball moving with velocity 40 m/s SE (-45 deg) in standard unit vector notation. Also, calculate the x- and y-components of the momentum.
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Transcribed Image Text:**Problem Statement:**
Write the momentum vector of a 10 kg ball moving with a velocity of 40 m/s Southeast (-45 degrees) in standard unit vector notation. Also, calculate the x- and y-components of the momentum.
**Solution:**
1. **Determine the Components of Velocity:**
- The velocity vector can be split into x and y components using trigonometry.
- Given angle: -45 degrees (Southeast direction).
- Velocity in x-direction, \( v_x = v \cdot \cos(\theta) \)
- \( v_x = 40 \, \text{m/s} \cdot \cos(-45^\circ) \)
- Velocity in y-direction, \( v_y = v \cdot \sin(\theta) \)
- \( v_y = 40 \, \text{m/s} \cdot \sin(-45^\circ) \)
2. **Calculate the Components:**
- \( \cos(-45^\circ) = \sin(-45^\circ) = \frac{\sqrt{2}}{2} \)
- \( v_x = 40 \, \text{m/s} \cdot \frac{\sqrt{2}}{2} = 20\sqrt{2} \, \text{m/s} \)
- \( v_y = 40 \, \text{m/s} \cdot \frac{\sqrt{2}}{2} = -20\sqrt{2} \, \text{m/s} \)
3. **Calculate Momentum Components:**
- Momentum, \( p = m \cdot v \)
- Mass, \( m = 10 \, \text{kg} \)
- Momentum in x-direction, \( p_x = m \cdot v_x \)
- \( p_x = 10 \, \text{kg} \cdot 20\sqrt{2} \, \text{m/s} = 200\sqrt{2} \, \text{kg} \cdot \text{m/s} \)
- Momentum in y-direction, \( p_y = m \cdot v_y \)
- \( p_y = 10 \, \text{kg} \cdot (-20\sqrt{2}) \, \text{m/s} = -200
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