Optimal Dragster Exhuast Pipe Angle: Energy of Systems and Force

[winowmak3r]

Homework Statement

I'm having a real hard time on the last homework question for this week for my Uni Physics I course. Here it is:

Top-fuel dragsters can accelerate from rest at a rate of about 5g. If this does not sound impressive, consider that for most automobiles the friction between the ground and the tires produces the acceleration. Since the coefficient of friction between regular treaded tires and pavement is typically less than or about equal to 1.0, a car relying on friction with normal force equal to its weight should accelerate at no more than about 1g.
Dragsters use two methods to increase their acceleration. First, they greatly increase the friction between their tires and the road by eliminating tire tread (dragsters do not race in wet or snowy road conditions) and also by performing a “burnout” which lays down a patch of heated rubber on the track that the partially-melted tires will adhere to. The result of these measures is a coefficient of friction exceeding 2.0. The second important method for increasing acceleration is the use of engine exhaust to provide force. The high-speed exhaust gases exiting the dragster’s enormous 8000 hp engine produce a force with a size comparable to the car’s weight.


a.) Assuming a coefficient of friction equal to 2.5, at what angle θ measured with respect to the horizontal (bottom picture) should the exhaust pipes be oriented in order to achieve maximum acceleration? Assume the engine is capable of using the tires to produce a force equal to the maximum friction force (μN) regardless of how large the normal force becomes!
θ
b.) Supposing the coefficient of friction is 2.5 and the exhaust is configured at the optimal angle found in part (a), how much force (as a fraction of the dragster’s weight) would the exhaust have to produce if the dragster accelerates from rest at 4.7g?

Homework Equations

F=m*a
F(friction)=uN
W=F*r*cos(theta)
More?

The Attempt at a Solution

I honestly have no idea. The correct answer for part a is 68 degrees (After much coercing the professor gave us the answer to part a but wouldn't tell us how to get there yet). A friend and I tried to do this problem and came up with 0 degrees but the logic was pretty shaky. Basically we reasoned that if u is 2.5, and the normal force is m*g*cos(theta) then you would want cos(theta) to be 1, hence zero degrees.

Any help is much appreciated!

 

[collinsmark]

Hello winowmak3r,

Oooh, this is a fun [phun] physics problem! 'Best I've seen in awhile.

a.) Assuming a coefficient of friction equal to 2.5, at what angle θ measured with respect to the horizontal (bottom picture) should the exhaust pipes be oriented in order to achieve maximum acceleration? Assume the engine is capable of using the tires to produce a force equal to the maximum friction force (μN) regardless of how large the normal force becomes!

[...]

The Attempt at a Solution

I honestly have no idea. The correct answer for part a is 68 degrees (After much coercing the professor gave us the answer to part a but wouldn't tell us how to get there yet). A friend and I tried to do this problem and came up with 0 degrees but the logic was pretty shaky.

It's not zero degrees. Your instructor's ~68^o answer to part a.) is correct.

Basically we reasoned that if u is 2.5, and the normal force is m*g*cos(theta) then you would want cos(theta) to be 1, hence zero degrees.

The normal force is not mgcosθ. The car is on a flat, horizontal surface. So the part of the normal force due to the car's weight is not a function of θ. In addition to the car's weight, there is the force of exhaust pushing down on the car, and that is what is a function of θ. Both the car's weight and this exhaust force combine to make up the car's normal force.

Start by defining a notation for the magnitude of the force of the exhaust. Call it F_e (you can call it anything you want, but you have to call it something). You don't know what its actual value is yet. And it turns out you don't need to know its value. But you do need to put F_e into your equations.

Now let me guide you by asking the following questions:

o What is the normal force (in terms of m, g, F_e, and θ)?
o How do you express Newton's second law (ma = ∑F) in the x-direction (horizontal only)? (Hint: there are two forces involve, the frictional force that is related to the normal force, and also the exhaust force which is related to F_e and θ.)
o What value of θ maximizes the acceleration a? (Hint: it involves taking the derivative of something and setting the result equal to zero. Answer pops right out [well, with a tiny bit of algebra, that is].)

Good luck!