The (non-conservative) force propelling a 1.90 x 103-kg car up a mountain road does 6.70 x 106 J of work on the car. The car starts from rest at sea level and has a speed of 20.0 m/s at an altitude of 1.80 x 102 m above sea level. Obtain the work done on the car by the combined forces of friction and air resistance, both of which are non-conservative forces.
To obtain the work done on the car by the combined forces of friction and air resistance, we need to determine the work done by the non-conservative force propelling the car up the mountain road, and subtract it from the total work done on the car.
Let's first calculate the work done by the non-conservative force propelling the car up the mountain road. We know that the work done by a force is given by the equation:
Work = Force x Distance x cos(θ)
Where:
- Work is the work done by the force in joules (J)
- Force is the magnitude of the force in newtons (N)
- Distance is the displacement in meters (m)
- θ (theta) is the angle between the direction of the force and the direction of displacement
In this case, the force propelling the car up the mountain road is the non-conservative force, the distance is the altitude gained, and the angle θ is 0 degrees (since the force is acting in the same direction as the displacement).
So, the work done by the non-conservative force is given by:
Work_nc = Force_nc x Distance x cos(θ)
The mass of the car is given as 1.90 x 103 kg, and the altitude gained is 1.80 x 102 m. However, the force is not directly given. We can find the force by using the work-energy theorem, which states that the work done by all forces equals the change in kinetic energy. In this case, the work done by the non-conservative force is equal to the change in kinetic energy of the car.
Given that the car starts from rest, the initial kinetic energy is zero (KE_initial = 0). The final kinetic energy can be calculated using the formula:
KE_final = (1/2)mv^2
Where:
- KE_final is the final kinetic energy
- m is the mass of the car
- v is the final velocity of the car
In this case, the mass of the car is 1.90 x 103 kg, and the final velocity is 20.0 m/s. Plugging in these values, we can calculate the final kinetic energy.
Next, we can use the work-energy theorem to find the work done by the non-conservative force:
Work_nc = KE_final - KE_initial
Now that we have the work done by the non-conservative force, we can substitute it into the equation to find the work done by the combined forces of friction and air resistance:
Work_friction_and_airresistance = Total Work - Work_nc
Given that the total work done on the car is 6.70 x 106 J, we can substitute these values into the equation to find the work done by the combined forces.