Car going up a hill. Find the Angle of the hill.

[pmd28]

Homework Statement

A 1550-kg car experiences a combined force of air resistance and friction that has the same magnitude whether the car goes up or down a hill at 21 m/s. Going up a hill, the car's engine produces 44 hp more power to sustain the constant velocity than it does going down the same hill. At what angle is the hill inclined above the horizontal?

Homework Equations

P=W/t=FvCosθ
W=fΔx

The Attempt at a Solution

I drew out my free body diagram and I have the Normal force, friction, and the components of the weight force (mgCosθ in the y-direction and mgSinθ in the x-direction)
I figured I need to find out the F in P=FvCosθ. I'm just not sure what F it is I have to find.

 

[TSny]

There's another force you need to include. What force "propels" the car up the hill?

 

[pmd28]

i thought wheels generated friction which caused a recipriocal force on the car by the road, which caused a car to move.

 

[TSny]

Exactly. I'm not sure that the "friction" mentioned in the problem is the same as the friction generated by the wheels to drive the car up the hill. I think they might have been talking about internal friction that acts along with the air resistance. But, anyway, I think we're clear on the forces.

Think of the force generated by the wheels as ultimately coming from the engine. So, if you are going to set up an expression for the power that the engine produces, you are going to need the force generated by the engine (i.e., the friction force generated by the wheels.) Can you think of a way to get an expression for that force?

 

[pmd28]

Well I'm given power and I'm velocity. Would it be right to use a rearrangement of P=Fv, F=P/v?

 

[TSny]

OK, but note they only give you the change in power when switching between going up and coming back down.

Can you bring in something else that would let you relate F to the angle of the slope?

 

[pmd28]

The components of the Weight force?

 

[TSny]

I was thinking more along the lines of a principle or law that you could apply. Note that they tell you the velocity is constant.

 

[pmd28]

Well if velocity is constant it can't be Newton's 2nd Law. I'm stumped.

 

[TSny]

Oh yes it can :)

 

[pmd28]

Well if it's at a constant velocity wouldn't that make F=0 due to the lack of acceleration?

 

[TSny]

But what does F represent in the 2nd law?

 

[pmd28]

Net Force?

 

[TSny]

Sure. That's important. F is always the sum of all of the forces acting on the object when setting up the 2nd law. Or, if you're working with components, it says that the sum of all the components of the forces in a certain direction equals the mass times the component of the acceleration in that direction.

 

[pmd28]

So do I use the component(s) of the Weight force to relate F and theta?

 

[TSny]

Let's pick a specific symbol for each force: f for the air resistance, mg for the weight, Fw for the force from the wheels, and Fn for the normal force. How would you write an expression for the net force acting upward along the slope?

 

[pmd28]

ƩF=Fw-f-mgcosθ

 

[TSny]

Good. Can you use the 2nd law to express Fw in terms of the other forces? [Oops: You didn't quite get that right. Think again about the component of the weight.]

 

[pmd28]

Yes, because ƩF=0, (because I know that the car is moving at a constant v), Fw = f + mgCosθ

 

[TSny]

Yes, but you'll need to correct the mgCos(theta) term.

 

[pmd28]

What do you mean?

 

[TSny]

mgCos(theta) is not the correct expression for the component of the weight that acts down the slope.

 

[pmd28]

oh, mgsin(theta). Yea I was reading my Free-Body Diagram wrong.

 

[TSny]

That's it. What would the result for Fw be for coming back down the slope.? (Drawing another diagram would be a good idea.)

 

[pmd28]

I'm not sure I understand that question.

 

[TSny]

The question gives you the difference in power of the engine as you switch from going down the slope to going up the slope. So, you'll need to think about how Fw changes when going down compared to up.

 

[pmd28]

Would it be right to plug in Fw into P=Fv?

 

[TSny]

You have found that Fw = f + mgSin(theta) going up.

Would the equation be the same for going back down? If not, how would it change? Just repeat the steps that you used for getting the equation for going up (including drawing a good free-body diagram).

 

[pmd28]

Wouldn't it be the same?

 

[TSny]

Do all the forces act in the same direction coming down as they did going up?