Certainly! To determine the magnitude of the change in momentum of the driver during a car accident, where a 125 kg driver is brought to rest from an initial velocity of [tex]\( 31 \, \text{m/s} \)[/tex], we need to follow these steps:
1. Understand Momentum and Its Change:
- Momentum ([tex]\( p \)[/tex]) of an object is given by the product of its mass ([tex]\( m \)[/tex]) and its velocity ([tex]\( v \)[/tex]): [tex]\( p = m \times v \)[/tex].
- Change in momentum ([tex]\( \Delta p \)[/tex]) is the difference between the final momentum and the initial momentum. Mathematically: [tex]\( \Delta p = m \times \Delta v \)[/tex], where [tex]\( \Delta v = v_{\text{final}} - v_{\text{initial}} \)[/tex].
2. Set Given Values:
- Mass of the driver ([tex]\( m \)[/tex]) = 125 kg
- Initial velocity ([tex]\( v_{\text{initial}} \)[/tex]) = 31 m/s
- Final velocity ([tex]\( v_{\text{final}} \)[/tex]) = 0 m/s (brought to rest)
3. Calculate Change in Velocity ([tex]\( \Delta v \)[/tex]):
[tex]\[
\Delta v = v_{\text{final}} - v_{\text{initial}} = 0 \, \text{m/s} - 31 \, \text{m/s} = -31 \, \text{m/s}
\][/tex]
4. Calculate Change in Momentum ([tex]\( \Delta p \)[/tex]):
[tex]\[
\Delta p = m \times \Delta v = 125 \, \text{kg} \times (-31 \, \text{m/s}) = -3875 \, \text{kg} \cdot \text{m/s}
\][/tex]
5. Determine Magnitude of Change in Momentum:
- The magnitude of a value is its absolute amount, regardless of sign. So, the magnitude of the change in momentum is:
[tex]\[
|\Delta p| = | -3875 \, \text{kg} \cdot \text{m/s} | = 3875 \, \text{kg} \cdot \text{m/s}
\][/tex]
6. Conclusion:
- Therefore, the magnitude of the change in momentum of the driver is [tex]\( 3875 \, \text{kg} \cdot \text{m/s} \)[/tex].
Given the provided options:
- [tex]\( 4.0 \, \text{kg} \cdot \text{m/s} \)[/tex]
- [tex]\( 21 \, \text{kg} \cdot \text{m/s} \)[/tex]
- [tex]\( 47 \, \text{kg} \cdot \text{m/s} \)[/tex]
- [tex]\( 3900 \, \text{kg} \cdot \text{m/s} \)[/tex]
The correct answer approximating our result [tex]\( 3875 \, \text{kg} \cdot \text{m/s} \)[/tex] is:
[tex]\[ \boxed{3900 \, \text{kg} \cdot \text{m/s}} \][/tex]
