To find the magnitude of the final momentum of both the car and the truck when they collide inelastically, we need to break it down step-by-step.
1. Determine the momentum of the car:
- Given:
- Mass of the car: [tex]\( m_{car} = 800 \)[/tex] kilograms
- Velocity of the car: [tex]\( v_{car} = 18 \)[/tex] meters/second (south)
- The momentum of the car (south direction) can be calculated using:
[tex]\[
p_{car} = m_{car} \times v_{car}
\][/tex]
[tex]\[
p_{car} = 800 \times 18 = 14400 \text{ kilogram meters/second}
\][/tex]
2. Determine the momentum of the truck:
- Given:
- Mass of the truck: [tex]\( m_{truck} = 1500 \)[/tex] kilograms
- Velocity of the truck: [tex]\( v_{truck} = 15 \)[/tex] meters/second (east)
- The momentum of the truck (east direction) can be calculated using:
[tex]\[
p_{truck} = m_{truck} \times v_{truck}
\][/tex]
[tex]\[
p_{truck} = 1500 \times 15 = 22500 \text{ kilogram meters/second}
\][/tex]
3. Calculate the magnitude of the final momentum:
- Since the car and truck are moving in perpendicular directions (south and east), their momentums form vectors at right angles to each other. The magnitude of the resultant momentum can be found using the Pythagorean theorem:
[tex]\[
p_{final} = \sqrt{(p_{car})^2 + (p_{truck})^2}
\][/tex]
[tex]\[
p_{final} = \sqrt{(14400)^2 + (22500)^2}
\][/tex]
[tex]\[
p_{final} = \sqrt{207360000 + 506250000}
\][/tex]
[tex]\[
p_{final} = \sqrt{713610000} \approx 26713.48 \text{ kilogram meters/second}
\][/tex]
Given the calculated magnitude of the final momentum, the answer closest to the value we obtained is:
D. [tex]\( 2.7 \times 10^4 \)[/tex] kilogram meters/second
