A mechanic pushes a 2.20 103-kg car from rest to a speed of v, doing 4,550 J of work in the process. During this time, the car moves 24.0 m. Neglecting friction between car and road, find each of the following.
Why did the mechanic push the car? Because the car forgot its keys and couldn't drive itself!
To find each of the following variables, we need to use the Work-Energy principle:
1. The final speed, v:
The work done on an object is equal to the change in its kinetic energy. So we can write the equation as:
Work = Change in Kinetic Energy
Given that the work done (W) is 4,550 J, and the car is initially at rest, the initial kinetic energy (KE_initial) is zero. Therefore, we can rewrite the equation as:
4,550 J = KE_final - KE_initial
Since the car starts from rest, KE_initial = 0. Therefore:
4,550 J = KE_final
We can use the equation for kinetic energy:
KE = (1/2) * m * v^2
Substituting the known values:
4,550 J = (1/2) * (2.20 * 10^3 kg) * v^2
Now, we can solve for v:
v^2 = (2 * 4,550 J) / (2.20 * 10^3 kg)
v^2 = 4145.45 J / (2.20 * 10^3 kg)
v^2 = 1.883 J/kg
v = √(1.883 J/kg)
v ≈ 1.37 m/s (rounded to two decimal places)
Therefore, the final speed of the car is approximately 1.37 m/s.
2. The work done against friction:
Since we are neglecting friction between the car and the road, the work done against friction is zero.
3. The average power expended by the mechanic:
Power is defined as the rate at which work is done. It can be calculated using the formula:
Power = Work / Time
Given that the work done (W) is 4,550 J, and the car travels a distance (d) of 24.0 m, we can calculate the time (t) using the formula:
t = d / v
t = 24.0 m / 1.37 m/s
t ≈ 17.52 s (rounded to two decimal places)
Now, we can calculate the average power:
Power = 4,550 J / 17.52 s
Power ≈ 260.15 W (rounded to two decimal places)
Therefore, the average power expended by the mechanic is approximately 260.15 W.