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Answer :
To solve the problem of finding the final velocity of the car, we can use the work-energy principle, which relates the work done on an object to its change in kinetic energy. Here's a step-by-step guide to solving the question:
1. Identify the Given Values:
- Mass of the car, [tex]\( m = 198 \text{ kg} \)[/tex].
- Initial velocity of the car, [tex]\( v_i = 12.3 \text{ m/s} \)[/tex].
- Force applied by the engine, [tex]\( F = 1146 \text{ N} \)[/tex].
- Distance over which the force is applied, [tex]\( d = 12.0 \text{ m} \)[/tex].
2. Calculate the Work Done:
The work done by the engine is calculated by multiplying the force by the distance over which the force acts:
[tex]\[
\text{Work Done} = F \times d = 1146 \times 12.0 = 13752 \text{ J}
\][/tex]
3. Calculate the Initial Kinetic Energy:
The initial kinetic energy is given by the formula:
[tex]\[
\text{Kinetic Energy}_{\text{initial}} = \frac{1}{2} m v_i^2
\][/tex]
Plugging in the values, we get:
[tex]\[
\text{Kinetic Energy}_{\text{initial}} = \frac{1}{2} \times 198 \times (12.3)^2 \approx 14977.71 \text{ J}
\][/tex]
4. Calculate the Final Kinetic Energy:
According to the work-energy principle, the work done on the car is equal to the change in its kinetic energy. Thus, the final kinetic energy is:
[tex]\[
\text{Kinetic Energy}_{\text{final}} = \text{Kinetic Energy}_{\text{initial}} + \text{Work Done}
\][/tex]
[tex]\[
\text{Kinetic Energy}_{\text{final}} = 14977.71 + 13752 = 28729.71 \text{ J}
\][/tex]
5. Solve for the Final Velocity:
The final kinetic energy is also given by the formula:
[tex]\[
\text{Kinetic Energy}_{\text{final}} = \frac{1}{2} m v_f^2
\][/tex]
Rearranging the formula to solve for the final velocity [tex]\( v_f \)[/tex], we have:
[tex]\[
v_f^2 = \frac{2 \times \text{Kinetic Energy}_{\text{final}}}{m}
\][/tex]
[tex]\[
v_f^2 = \frac{2 \times 28729.71}{198}
\][/tex]
[tex]\[
v_f \approx \sqrt{290.49} \approx 17.04 \text{ m/s}
\][/tex]
Therefore, the final velocity of the car is approximately [tex]\( 17.04 \text{ m/s} \)[/tex].
1. Identify the Given Values:
- Mass of the car, [tex]\( m = 198 \text{ kg} \)[/tex].
- Initial velocity of the car, [tex]\( v_i = 12.3 \text{ m/s} \)[/tex].
- Force applied by the engine, [tex]\( F = 1146 \text{ N} \)[/tex].
- Distance over which the force is applied, [tex]\( d = 12.0 \text{ m} \)[/tex].
2. Calculate the Work Done:
The work done by the engine is calculated by multiplying the force by the distance over which the force acts:
[tex]\[
\text{Work Done} = F \times d = 1146 \times 12.0 = 13752 \text{ J}
\][/tex]
3. Calculate the Initial Kinetic Energy:
The initial kinetic energy is given by the formula:
[tex]\[
\text{Kinetic Energy}_{\text{initial}} = \frac{1}{2} m v_i^2
\][/tex]
Plugging in the values, we get:
[tex]\[
\text{Kinetic Energy}_{\text{initial}} = \frac{1}{2} \times 198 \times (12.3)^2 \approx 14977.71 \text{ J}
\][/tex]
4. Calculate the Final Kinetic Energy:
According to the work-energy principle, the work done on the car is equal to the change in its kinetic energy. Thus, the final kinetic energy is:
[tex]\[
\text{Kinetic Energy}_{\text{final}} = \text{Kinetic Energy}_{\text{initial}} + \text{Work Done}
\][/tex]
[tex]\[
\text{Kinetic Energy}_{\text{final}} = 14977.71 + 13752 = 28729.71 \text{ J}
\][/tex]
5. Solve for the Final Velocity:
The final kinetic energy is also given by the formula:
[tex]\[
\text{Kinetic Energy}_{\text{final}} = \frac{1}{2} m v_f^2
\][/tex]
Rearranging the formula to solve for the final velocity [tex]\( v_f \)[/tex], we have:
[tex]\[
v_f^2 = \frac{2 \times \text{Kinetic Energy}_{\text{final}}}{m}
\][/tex]
[tex]\[
v_f^2 = \frac{2 \times 28729.71}{198}
\][/tex]
[tex]\[
v_f \approx \sqrt{290.49} \approx 17.04 \text{ m/s}
\][/tex]
Therefore, the final velocity of the car is approximately [tex]\( 17.04 \text{ m/s} \)[/tex].
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