Kinematics - Constant Acceleration

[JG89]

Homework Statement

To demonstrate the tremendous acceleration of a top fuel drag racer, you attempt to run your car into the back of a dragster that is "burning out" at the red light before the start of a race. (Burning out means spinning the tires at high speed to heat the tread and make the rubber sticky.)

You drive at a constant speed of [tex] v_0 [/tex] toward the stopped dragster, not slowing down in the face of the imminent collision. The dragster driver sees you coming but waits until the last instant to put down the hammer, accelerating from the starting line at constant acceleration, a. Let the time at which the dragster starts to accelerate be t=0.

What is t_max, the longest time after the dragster begins to accelerate that you can possibly run into the back of the dragster if you continue at your initial velocity?

Homework Equations

The Attempt at a Solution

I have three motion equations for constant acceleration. But I have no idea where to start this problem...I know it's not going to be a numerical answer. The answer will be expressed in terms of my variables.

 

[TVP45]

Is there a question in there?

 

[JG89]

Oops. I just edited my original post with the question included.

 

[TVP45]

Are there any actual numbers in there?

 

[JG89]

Nope

 

[TVP45]

Not even for "last instant"? What do you suppose that means?

 

[JG89]

Not even for "last instant"...

I suppose that means that my answer will be expressed in variables...If the drag-racer waits until the last instant to hit the pedal, then shouldn't my car and the race-car's position be virtually the same at t = 0?

If at t-max our positions are again the same, then it seems like I am going to have to use the equation [tex] s_f = s_i + v_i(t_{max} - t_0) + 0.5a(t_{max} - t_0)^2 [/tex], as s_f will have two solutions (one at t = 0 and one at t-max) and I will probably be solving a quadratic...

 

[JG89]

Okay, I think I've got it.

For my car: [tex] s_f = s_i + v_i(t_{max} - t_0) [/tex]. The question said to put t_0 = 0, and I also know that s_i = 0, so the equation simplifies to [tex] s_f = v_i(t_{max}) [/tex], where v_i is the constant velocity that I'm traveling at (it's not 0).

Now, I want to know when the drag car is also at position s_f. So the equation for the drag-car is: [tex] s_f = s_i + v_{i-drag}(t_{max} - t_0) + 0.5a(t_{max} - t_i)^2 [/tex].
v_i-drag is the drag-cars initial velocity, which is 0. Also remember that t_i = 0. s_i also equals 0. So the equation simplifies to [tex] s_f = 0.5a(t_{max})^2 [/tex].

Setting the two equations equal to each other: [tex] 0.5a(t_{max})^2 = v_i(t_{max}) \Leftrightarrow -0.5a(t_{max})^2 + v_i(t_{max}) = 0 [/tex]. Using the quadratic formula, I have [tex] t_{max} = \frac{-v_i \pm v_i}{-a} [/tex]. There are two solutions to this. When [tex] t_{max} = 0 [/tex] and when [tex] t_{max} = \frac{2v_i}{a} [/tex]. I will take the second solution.

Is this correct?

 

[TVP45]

I suppose you might state "last instant' as referring to some distance before the eminent collision (and that could as easily be some time), so you might decide that the "chase car" is traveling at V_o and the dragster accelerates when the chase car is -x behind it. The quadratic implies you would be coincident twice and that can't happen if you're on a single track. So, I think you have to consider the first collision and you might as well let that happen down at the traps.

 

[JG89]

So then can't I take my original answer of t_max = 2v_i/a and just subtract a tiny bit of time from it? So t_max = 2v_i/a - t, where t is a small bit of time?

 

[TVP45]

For some reason, I am unable to read your equations.

However, I am a bit puzzled by the ambiguity of the question. If someone else sees a "best assumption", and can jump in, that might be better. Here's where I'm getting puzzled:

If the two cars are coincident, and V_car > a_dragstert, they have collided. Obviously, the cars cannot be coincident at the start. Thus, there is a distance between them at the start, which you can express as -X_o-car or as -V_cart_startline, where t_startline is the time between t₀ and the time the car crosses the drag race start line. Now, you can easily set the two distances equal, as you seem to have done, and solve for t. You likely get two roots (This is easy to see with 2 tilted lab tracks and a constant velocity car on one and a free wheeling car on the other), but only the smaller is of use.

But, and this is my puzzle, that is only t, not t_max. Clearly, if I make the dragster acceleration approach 0, and set the velocity of the car to just collide at the very end of the drag strip, that is t_max. Yet, that seems to run against the idea of showing the tremendous acceleration of a top fuel dragster. I'm sure I'm being too literal, yet I can't see any way out of this, except perhaps partial derivatives?

I could use some help from someone who is not so literal as I.