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A 198 kg car on flat ground is initially moving at [tex]12.3 \, \text{m/s}[/tex]. The engine of the car creates a 1146 N forward force as it moves 12.0 m. What is the final velocity of the car?

[tex]v_f = [?] \, \text{m/s}[/tex]

Answer :

To find the final velocity of the car, we can use the work-energy principle, which states that the work done on an object is equal to the change in its kinetic energy. Here’s a step-by-step solution:

1. Identify Given Values:
- Mass of the car, [tex]\( m = 198 \, \text{kg} \)[/tex]
- Initial velocity, [tex]\( v_i = 12.3 \, \text{m/s} \)[/tex]
- Force applied, [tex]\( F = 1146 \, \text{N} \)[/tex]
- Distance over which the force is applied, [tex]\( d = 12.0 \, \text{m} \)[/tex]

2. Calculate Work Done:
- Work done by the force is calculated using the formula:
[tex]\[
\text{Work} = F \times d
\][/tex]
Substituting the values, we have:
[tex]\[
\text{Work} = 1146 \, \text{N} \times 12.0 \, \text{m} = 13752 \, \text{J}
\][/tex]

3. Calculate Initial Kinetic Energy:
- The initial kinetic energy ([tex]\( KE_i \)[/tex]) of the car is given by:
[tex]\[
KE_i = \frac{1}{2} m v_i^2
\][/tex]
Substituting the given values:
[tex]\[
KE_i = \frac{1}{2} \times 198 \, \text{kg} \times (12.3 \, \text{m/s})^2 = 14977.71 \, \text{J}
\][/tex]

4. Calculate Final Kinetic Energy:
- The final kinetic energy ([tex]\( KE_f \)[/tex]) is the sum of the initial kinetic energy and the work done:
[tex]\[
KE_f = KE_i + \text{Work}
\][/tex]
Substituting the calculated values:
[tex]\[
KE_f = 14977.71 \, \text{J} + 13752 \, \text{J} = 28729.71 \, \text{J}
\][/tex]

5. Calculate Final Velocity:
- The final velocity ([tex]\( v_f \)[/tex]) can be found using the final kinetic energy formula:
[tex]\[
KE_f = \frac{1}{2} m v_f^2
\][/tex]
Solving for [tex]\( v_f \)[/tex]:
[tex]\[
v_f = \sqrt{\frac{2 \times KE_f}{m}}
\][/tex]
Substituting the known values:
[tex]\[
v_f = \sqrt{\frac{2 \times 28729.71 \, \text{J}}{198 \, \text{kg}}} = 17.035 \, \text{m/s}
\][/tex]

The final velocity of the car is approximately [tex]\( 17.04 \, \text{m/s} \)[/tex].

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