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Chapter 6: Problem 29

At what speed must a 950 -kg subcompact car be moving to have the same kinetic energy as a \(3.2 \times 10^{4}-\mathrm{kg}\) truck going \(20 \mathrm{km} / \mathrm{h} ?\)

Short Answer

Expert verified
The speed of the subcompact car must be approximately 32.97 m/s to have the same kinetic energy as the truck traveling at 20 km/hr.

Step by step solution

01

Conversion of Speed units for the Truck

Firstly, convert the speed of the truck from kilometers per hour (km/h) to meters per second (m/s). This is essential because the standard unit for speed in physics is m/s. The conversion factor is 1 km/h equals 0.27778 m/s. Therefore, \(20 \, km/h = 20 \times 0.27778 \, m/s = 5.56 \, m/s.\)
02

Calculation of Truck's Kinetic Energy

Use the equation for Kinetic energy (\(KE = \frac{1}{2} m v^{2}\)) to calculate the truck's kinetic energy substituting its mass and velocity as follows: \(KE_{Truck} = \frac{1}{2} \times 3.2 \times 10^{4} \, kg \times (5.56 \, m/s)^{2}\). This leads to \(KE_{Truck} \approx 500,000 \, Joules\).
03

Calculate the Speed of Subcompact car

The question's crux is finding the speed of the car that results in this same kinetic energy. The equation for kinetic energy is rearranged to solve for the car's velocity. So, \(v = \sqrt{\frac{2KE}{m}}\), substituting the car's mass and the truck's kinetic energy we find, \(v= \sqrt{\frac{2 \times 500,000 \, Joules}{950 \, kg}}\).

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