# Max acceleration of car up ramp

#### diffusion

1. Homework Statement
A car is driving up a slope at angle 16 deg. to the horizontal, trying to accelerate as much as possible. The static and kinetic friction coefficients are .90 and .80, respectively. Find the maximum possible acceleration (assuming a sufficiently powerful engine).

2. Homework Equations
Fs,max = .90 x normal force
Max acceleration = -mg(sin16) + fs,max / mass

3. The Attempt at a Solution
Ok, I've seen a problem similar to this one and (plugging in the values from this question) it was solved using this first step:

Max acceleration = -mg(sin16) + fs,max / mass.

My first question is, where did the (sin16) come from? I drew a free body force diagram to look like this:

EDIT: Sorry, in the image, f,static should actually be f,kinetic, since the car is accelerating.

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#### sArGe99

Doesn't that depend on how much power the engine can generate?

#### diffusion

Doesn't that depend on how much power the engine can generate?
The question says "Assuming a sufficiently powerful engine", so I'm guessing... no?

#### diffusion

Anyone? Still have no clue where the sin16 came from. Opposite over hypotenuse, I don't see it.

#### Doc Al

Mentor
Still have no clue where the sin16 came from.
If you break the weight (mg, down) into components, the component parallel to the incline will be mg sinθ (down the incline). (If you think in terms of a right triangle, mg will be the hypotenuse.)

#### diffusion

If you break the weight (mg, down) into components, the component parallel to the incline will be mg sinθ (down the incline). (If you think in terms of a right triangle, mg will be the hypotenuse.)
I feel like I'm never going to grasp this stuff. I just cannot see where you're getting this from.

Could you perhaps draw a diagram of what you mean? If not, thanks anyway.

#### Doc Al

Mentor
Study this: http://www.physicsclassroom.com/Class/vectors/u3l3e.cfm" [Broken]

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#### diffusion

Study this: http://www.physicsclassroom.com/Class/vectors/u3l3e.cfm" [Broken]
That was extremely helpful, thanks so much. I have 3 physics textbooks and none of them seem to explain that very well at all.

I'll give the question another shot now on my own and see what I get.

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#### diffusion

Ok, I understand what I'm doing now (I think), but I can only seem to solve for max acceleration in terms of n or m. Can't seem to get a direct answer without knowing the mass of the car. Here's my work:

1. Separate the components of mg.

2. To find the maximum acceleration in the presence of friction, use the equation a$$_{max}$$ = mg(sin$$\theta$$) + f$$_{s,max}$$ / m.

3. We're only given the coefficient of static friction, so to find f$$_{s,max}$$, we use f$$_{s,max}$$ = $$\mu$$$$_{s}$$ x n.

4. Plugging in the coefficient of static friction we get f$$_{s,max}$$ = (.90)n

5. Plug into original equation: a$$_{max}$$ = mg(sin16) + (.90)n / m.

6. Mass cancels, plug in g: a$$_{max}$$ = 9.8(sin16) + (.90)n.

7. Simplify: a$$_{max}$$ = 2.70 + (.90)n.

That's as far as I can go, don't know how to get rid of n without knowing the mass of the car. Or am I doing something wrong?

#### Doc Al

Mentor
Ok, I understand what I'm doing now (I think), but I can only seem to solve for max acceleration in terms of n or m. Can't seem to get a direct answer without knowing the mass of the car.
You don't need the mass to find the acceleration. It cancels out. Hint: Express the normal force in terms of mg.

Here's my work:

1. Separate the components of mg.
You need both components of mg, parallel and perpendicular to the ramp. The perpendicular component will tell you the normal force.

#### diffusion

You don't need the mass to find the acceleration. It cancels out. Hint: Express the normal force in terms of mg.

You need both components of mg, parallel and perpendicular to the ramp. The perpendicular component will tell you the normal force.
Right. The perpendicular component equals the normal force because they have to balance each other out. The equation for the perpendicular component is mg(cos16), still leaving me with an extra m. Should there be two m's in the denominator, say, m$$_{1}$$ + m$$_{2}$$, one for each component?

#### Doc Al

Mentor
Right. The perpendicular component equals the normal force because they have to balance each other out. The equation for the perpendicular component is mg(cos16), still leaving me with an extra m.
It's not "extra". It's just what you need so that the m's cancel nicely.
Should there be two m's in the denominator, say, m$$_{1}$$ + m$$_{2}$$, one for each component?

#### diffusion

Ok, so we have:

a$$_{max}$$ = mg(sin16) + (.90)mg(cos16) / m

Which can be factored into:

a$$_{max}$$ = m(g[sin16] + .90g[cos16]) / m

m's cancel, and plug in 9.8 for g:

a$$_{max}$$ = (9.8[sin16] + .90(9.8)[cos16])

a$$_{max}$$ = 11.17m/s$$^{2}$$

#### Doc Al

Mentor
Ok, so we have:

a$$_{max}$$ = mg(sin16) + (.90)mg(cos16) / m
You left out a minus sign in from of the mg(sin16). (Otherwise: Good!)

#### diffusion

You left out a minus sign in from of the mg(sin16). (Otherwise: Good!)
Oh, so rather, it would look like:

(-9.8[sin16] + .90(9.8)[cos16])

= 5.78 m/s$$_{2}$$

Mentor
Good!

#### xplosiv

Hey Doc Al, shouldn't the force diagram have static friction going UP the hill. Because the static friction is what drives cars?

#### Doc Al

Mentor
Hey Doc Al, shouldn't the force diagram have static friction going UP the hill. Because the static friction is what drives cars?
Absolutely. The diagram is wrong. I meant to point that out, but forgot. Thanks!

(But the equation ended up OK.)

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