A 76.3-kN car is travelling at 66.5 mph when the driver decides to exit the freeway by going up a ramp. After coasting 418 m along the exit ramp the car\'s speed is 27.3 mph, and it is h = 10.7 m above the freeway. What is the magnitude of the average drag force exerted on the car?
mg=76300 N, m=76300/9.8 kg, s= 418 m,
v₀=66.5 mph =29.7 m/s
v=28.9 mph=12.2 m/s, h=10.7 m
KE1-KE2= W(dr)+ PE
m•v₀²/2 - m•v²/2= F(dr) •s +m•g•h
Solve for “F(dr)”
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Well, it seems like our car is taking quite the detour! Let's see if we can navigate through this scientific maze using a little bit of humor.
First things first, we need to find the change in kinetic energy of the car. Why? Because energy is like that one friend who's always accounting for changes!
So, the initial kinetic energy can be calculated using the old formula: KE = (1/2) * mass * velocity^2. However, we need to convert the speeds from mph to m/s because science prefers metric measurements.
So, the initial velocity will be (66.5 mph) * (0.447 m/s per mph). And the final velocity will be (27.3 mph) * (0.447 m/s per mph). Voila!
Now, let's calculate the mass of the car. We're given the weight, but the car's mass is like its personal trainer – it's a little bit more concealed. We can find it by dividing the weight (76.3 kN) by the acceleration due to gravity (about 9.8 m/s^2). That will give us the mass in kg.
Once we have the initial and final kinetic energies, we can subtract them to find the change in kinetic energy.
Now, hold on tight, because here comes the fun part!
The change in kinetic energy is equal to the work done by the drag force. And the work done is equal to the force times the distance traveled. In this case, the force we're looking for is the average drag force.
Divide the change in kinetic energy by the 418 m to find the average drag force. Remember to keep your units consistent, or else physics will play a little joke on you!
So, that's how you calculate the average drag force exerted on the car. But remember, numbers alone are quite serious, so make sure to add some humor while tackling these scientific challenges!
To find the magnitude of the average drag force exerted on the car, we can use the work-energy principle.
First, let's convert the speeds from mph to m/s.
Given:
Initial speed (v1) = 66.5 mph
Final speed (v2) = 27.3 mph
1 mph = 0.44704 m/s
v1 = 66.5 mph * 0.44704 m/s/mph = 29.67848 m/s
v2 = 27.3 mph * 0.44704 m/s/mph = 12.215392 m/s
Next, let's calculate the change in kinetic energy (ΔKE) of the car.
ΔKE = KE2 - KE1
ΔKE = (1/2) * m * v2^2 - (1/2) * m * v1^2
Where:
m is the mass of the car (unknown)
v1 is the initial velocity
v2 is the final velocity
To find the mass of the car, we can use Newton's second law of motion:
ΣF = m * a
The only force acting in the vertical direction is the weight of the car, so we can write:
ΣF = m * g
Where:
ΣF is the net force in the vertical direction
m is the mass of the car
g is the acceleration due to gravity (approximately 9.8 m/s^2)
We know the weight of the car (W), which is given by:
W = m * g
Now, let's calculate the work done by the drag force.
The work done by the drag force is equal to the change in kinetic energy. Therefore,
Work = ΔKE
Finally, we can find the magnitude of the average drag force by dividing the work done by the displacement along the ramp.
Drag force = Work / Displacement
Given:
Displacement (s) = 418 m
Change in kinetic energy (ΔKE) = Work (from above calculations)
Let's calculate the magnitude of the average drag force exerted on the car step by step.
To find the magnitude of the average drag force exerted on the car, we need to calculate the change in kinetic energy of the car. The change in kinetic energy is equal to the work done by the drag force.
Here's how you can calculate it step by step:
1. Convert the given values to SI units:
- Car's weight (W) = 76.3 kN = 76.3 × 1000 N = 76300 N
- Initial velocity (v1) = 66.5 mph = (66.5 × 1609) / 3600 m/s = 29.77 m/s
- Final velocity (v2) = 27.3 mph = (27.3 × 1609) / 3600 m/s = 12.21 m/s
- Distance traveled along the exit ramp (d) = 418 m
- Height above the freeway (h) = 10.7 m
2. Calculate the initial kinetic energy of the car:
- Initial kinetic energy (KE1) = (1/2) × mass × initial velocity^2
We can rearrange the formula to solve for mass:
mass = weight / acceleration due to gravity (g)
- acceleration due to gravity (g) = 9.8 m/s^2 (approximately)
- mass = 76300 N / 9.8 m/s^2
Now we can calculate the initial kinetic energy:
KE1 = (1/2) × mass × initial velocity^2
3. Calculate the final kinetic energy of the car:
- Final kinetic energy (KE2) = (1/2) × mass × final velocity^2
4. Calculate the change in kinetic energy:
- Change in kinetic energy (ΔKE) = KE2 - KE1
5. Calculate the work done by the drag force:
- Work = force × distance
- Force (F) = ΔKE / distance
6. Calculate the magnitude of the average drag force:
- Magnitude of the average drag force = |F|
By following these steps and plugging in the values given in the question, you can find the magnitude of the average drag force exerted on the car.