A 76.3-kN car is travelling at 68.7 mph when the driver decides to exit the freeway by going up a ramp. After coasting 355 m along the exit ramp the car\'s speed is 30.1 mph, and it is h = 14.3 m above the freeway. What is the magnitude of the average drag force exerted on the car?

To find the magnitude of the average drag force exerted on the car, we can use the work-energy principle.

Step 1: Convert the masses and speeds to SI units:
- Convert the force from kilonewtons to newtons: 76.3 kN = 76.3 × 1000 N = 76300 N.
- Convert the initial speed from mph to m/s: 68.7 mph = 68.7 × 0.44704 = 30.71253808 m/s.
- Convert the final speed from mph to m/s: 30.1 mph = 30.1 × 0.44704 = 13.455104 m/s.

Step 2: Calculate the initial kinetic energy (KE_initial) and final kinetic energy (KE_final) of the car:
- KE_initial = (1/2) × mass × initial_speed²

Step 3: Calculate the gravitational potential energy (PE_gravitational) of the car:
- PE_gravitational = mass × g × height

Step 4: Calculate the work done by the drag force (W_drag):
- W_drag = KE_initial - KE_final - PE_gravitational

Step 5: Calculate the magnitude of the average drag force (F_drag):
- F_drag = W_drag / distance

Now, let's calculate each value step by step:

Step 1: Conversion
- Initial force = 76.3 kN = 76.3 × 1000 N = 76300 N
- Initial speed = 68.7 mph = 68.7 × 0.44704 = 30.71253808 m/s
- Final speed = 30.1 mph = 30.1 × 0.44704 = 13.455104 m/s

Step 2: Calculate KE_initial
- KE_initial = (1/2) × mass × initial_speed²

Step 3: Calculate PE_gravitational
- PE_gravitational = mass × g × height

Step 4: Calculate W_drag
- W_drag = KE_initial - KE_final - PE_gravitational

Step 5: Calculate F_drag
- F_drag = W_drag / distance

Let's calculate each step:

Step 2: Calculate KE_initial
- KE_initial = (1/2) × mass × initial_speed²

Since we don't have the mass of the car, we cannot calculate the initial kinetic energy.

Step 3: Calculate PE_gravitational:
- PE_gravitational = mass × g × height

Since we don't have the mass of the car, we cannot calculate the gravitational potential energy.

Therefore, we cannot calculate the work done by the drag force (W_drag) or the magnitude of the average drag force (F_drag) without the mass of the car.

To find the magnitude of the average drag force exerted on the car, we can use the work-energy principle. The work done by the drag force is equal to the change in the car's kinetic energy.

First, we need to find the initial and final kinetic energy of the car.

Given:
Mass of the car (m) = 76.3 kN = 76300 N (since 1 kN = 1000 N)
Initial velocity of the car (v1) = 68.7 mph
Final velocity of the car (v2) = 30.1 mph

We need to convert mph (miles per hour) to m/s (meters per second) because energy is measured in SI units.

Converting velocities:
Initial velocity (v1) = 68.7 mph = 30.7488 m/s (1 mph ≈ 0.44704 m/s)
Final velocity (v2) = 30.1 mph = 13.44016 m/s

Now, we can calculate the initial and final kinetic energy using the formula:

Kinetic energy (K) = (1/2) * mass * velocity^2

Initial kinetic energy (K1) = (1/2) * m * v1^2
Final kinetic energy (K2) = (1/2) * m * v2^2

Plugging in the values:
K1 = (1/2) * 76300 * (30.7488)^2
K2 = (1/2) * 76300 * (13.44016)^2

Next, we need to find the work done by the drag force (W) using the formula:

W = K2 - K1

Substituting the values:
W = (1/2) * 76300 * (13.44016)^2 - (1/2) * 76300 * (30.7488)^2

Finally, to find the magnitude of the average drag force (F), we divide the work by the distance the car travels along the exit ramp (s).

Average drag force (F) = W / s

Given:
Distance travelled (s) = 355 m

Substituting the values:
F = [ (1/2) * 76300 * (13.44016)^2 - (1/2) * 76300 * (30.7488)^2 ] / 355

Simplifying the expression will give us the magnitude of the average drag force exerted on the car.