A 1260-kg car starts from rest at the bottom of a drive way and has a speed of 3.00 m/s at a point where the drive way has risen a vertical height of 0.600 m. Friction and the drive force produced by the engine are the only two nonconservative forces present. Friction does -2870 J of work. How much work does the engine do?
W=PE+KE+W(fr) =mgh+mv²/2 +W(fr) =
=m(gh+ v²/2)+W(fr) =1260(0.6•9.8 +3²/2)+2870 =
=13078.8+2870 =15948.8 J
To find the work done by the engine, we need to calculate the total work done, subtracting the work done by friction.
The work done by the engine can be determined using the work-energy theorem,
which states that the work done on an object is equal to the change in its kinetic energy.
We can start by finding the initial kinetic energy of the car:
Initial kinetic energy = 1/2 * mass * (initial velocity)^2
Given:
Mass (m) = 1260 kg
Initial velocity (v initial) = 0 m/s (car starts from rest)
Initial kinetic energy = 1/2 * 1260 kg * (0 m/s)^2 = 0 J
Next, we determine the final kinetic energy of the car:
Final kinetic energy = 1/2 * mass * (final velocity)^2
Given:
Final velocity (v final) = 3.00 m/s
Final kinetic energy = 1/2 * 1260 kg * (3.00 m/s)^2 = 5670 J
The total work done is equal to the change in kinetic energy:
Total work done = Final kinetic energy - Initial kinetic energy = 5670 J - 0 J = 5670 J
Now, to find the work done by friction, we know that it does -2870 J of work (negative because it opposes the motion). Therefore, the work done by the engine is:
Work done by the engine = Total work done - Work done by friction
Work done by the engine = 5670 J - (-2870 J) = 5670 J + 2870 J = 8540 J
Therefore, the work done by the engine is 8540 J.
To find the work done by the engine, we need to understand the concept of work and the work-energy principle.
Work is defined as the product of force and displacement in the direction of the force applied. Mathematically, work is given by the equation:
Work = Force * Displacement * cos(θ)
where θ is the angle between the force vector and the displacement vector.
The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. Mathematically, this can be expressed as:
Work done = Change in kinetic energy
In this case, we know that the only two forces acting on the car are friction and the drive force produced by the engine. Friction does -2870 J of work, which means it takes away energy from the car. The negative sign indicates that the work done by friction is in the opposite direction of the displacement.
We can calculate the work done by friction as:
Work done by friction = Friction force * Displacement * cos(180°)
Since the friction force and the displacement are in opposite directions (180° apart), the angle between them is 180°. Therefore, the cosine of 180° is -1, and we can simplify the equation as:
Work done by friction = Friction force * Displacement * -1
Given that the work done by friction is -2870 J and the vertical displacement is 0.600 m, we can rearrange the equation to find the friction force:
Friction force = Work done by friction / (Displacement * -1)
Friction force = -2870 J / (0.600 m * -1)
Friction force = 4783.33 N (rounded to two decimal places)
Now, we can find the work done by the engine using the work-energy principle. The total work done on the car is the sum of the work done by friction and the work done by the engine:
Total work done = Work done by friction + Work done by the engine
Since we know the change in kinetic energy is equal to the total work done, we can write:
Change in kinetic energy = Total work done
The initial kinetic energy of the car is zero since it starts from rest. The final kinetic energy can be calculated using the equation:
Final kinetic energy = (1/2) * mass * velocity^2
Given that the car has a mass of 1260 kg and a final velocity of 3.00 m/s, we can calculate the final kinetic energy:
Final kinetic energy = (1/2) * 1260 kg * (3.00 m/s)^2
Final kinetic energy = 5670 J
Now, substituting the values into the equation for the work-energy principle, we have:
Change in kinetic energy = Total work done
Final kinetic energy - Initial kinetic energy = Work done by friction + Work done by the engine
5670 J - 0 J = -2870 J + Work done by the engine
Work done by the engine = Final kinetic energy - Work done by friction
Work done by the engine = 5670 J - (-2870 J)
Work done by the engine = 5670 J + 2870 J
Work done by the engine = 8540 J
Therefore, the work done by the engine is 8540 J.