To find the difference in the automobile's kinetic energy between two velocities, let's perform the necessary calculations step-by-step.
First, we know the formula for kinetic energy ([tex]\( KE \)[/tex]) is:
[tex]\[ KE = \frac{1}{2} m v^2 \][/tex]
where
- [tex]\( m \)[/tex] is the mass of the object,
- [tex]\( v \)[/tex] is the velocity of the object.
Given data:
- The mass of the automobile ([tex]\( m \)[/tex]) is 450 kilograms.
- The initial velocity ([tex]\( v \)[/tex]) is 26 meters per second.
- The final velocity ([tex]\( v \)[/tex]) is 30 meters per second.
1. Calculate the initial kinetic energy:
[tex]\[
KE_{\text{initial}} = \frac{1}{2} \times 450 \times (26)^2
\][/tex]
Simplifying,
[tex]\[
KE_{\text{initial}} = 0.5 \times 450 \times 676
\][/tex]
[tex]\[
KE_{\text{initial}} = 0.5 \times 304200
\][/tex]
[tex]\[
KE_{\text{initial}} = 152100 \, \text{joules}
\][/tex]
2. Calculate the final kinetic energy:
[tex]\[
KE_{\text{final}} = \frac{1}{2} \times 450 \times (30)^2
\][/tex]
Simplifying,
[tex]\[
KE_{\text{final}} = 0.5 \times 450 \times 900
\][/tex]
[tex]\[
KE_{\text{final}} = 0.5 \times 405000
\][/tex]
[tex]\[
KE_{\text{final}} = 202500 \, \text{joules}
\][/tex]
3. Calculate the difference in kinetic energy:
[tex]\[
\Delta KE = KE_{\text{final}} - KE_{\text{initial}}
\][/tex]
[tex]\[
\Delta KE = 202500 - 152100
\][/tex]
[tex]\[
\Delta KE = 50400 \, \text{joules}
\][/tex]
Thus, the difference in the automobile's kinetic energy between the two velocities is [tex]\( \boxed{50400} \)[/tex] joules.
