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Calculate the difference in the automobile's kinetic energy between the two velocities. You may use a calculator.

Given:
[tex] K E = \frac{1}{2} m v^2 \]

A person drives an automobile with a mass of 450 kilograms at a velocity of 26 meters per second. The driver accelerates to a velocity of 30 meters per second. The difference in the automobile's kinetic energy between the two velocities is [tex]$\square$[/tex] joules.


Sagot :

To determine the difference in the automobile's kinetic energy between the two velocities, we will follow these steps:

1. Identify the given values:
- Mass of the automobile, [tex]\( m \)[/tex]: 450 kilograms
- Initial velocity, [tex]\( v_i \)[/tex]: 26 meters per second
- Final velocity, [tex]\( v_f \)[/tex]: 30 meters per second

2. Calculate the initial kinetic energy:
The kinetic energy (KE) of an object is given by the formula:
[tex]\[ KE = \frac{1}{2} m v^2 \][/tex]
Substituting the initial velocity [tex]\( v_i \)[/tex]:
[tex]\[ KE_{initial} = \frac{1}{2} \times 450 \, \text{kg} \times (26 \, \text{m/s})^2 \][/tex]

From the calculations:
[tex]\[ KE_{initial} = 152100 \, \text{joules} \][/tex]

3. Calculate the final kinetic energy:
Using the same formula but substituting the final velocity [tex]\( v_f \)[/tex]:
[tex]\[ KE_{final} = \frac{1}{2} \times 450 \, \text{kg} \times (30 \, \text{m/s})^2 \][/tex]

From the calculations:
[tex]\[ KE_{final} = 202500 \, \text{joules} \][/tex]

4. Calculate the difference in kinetic energy:
The difference in kinetic energy is the final kinetic energy minus the initial kinetic energy:
[tex]\[ \Delta KE = KE_{final} - KE_{initial} \][/tex]
Substituting the computed values:
[tex]\[ \Delta KE = 202500 \, \text{joules} - 152100 \, \text{joules} \][/tex]

The result is:
[tex]\[ \Delta KE = 50400 \, \text{joules} \][/tex]

Thus, the difference in the automobile's kinetic energy between the two velocities is [tex]\( 50400 \)[/tex] joules.