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.
