Solving AP Mechanics Problem: Find DeltaTy for Vo & Minimal Time Interval

[fantolay]

Homework Statement

Assume you are driving at the speed limit V_o. As you approach an intersection 22m wide, you see the light turn yellow. During your reaction time of .6s, you travel at a constant speed as you recognize the warning, decide whether to stop or to go through the intersection, and move your foot to the brake if you must stop. Your car has good brakes and can accelerate at -2.4m/s². Before it turns red, the light should stay yellow long enough for you to be able to get to the other side of the intersection without speeding up, if you are too close to the intersection to stop before entering it. (Suggestion: You may find it easier to do part (a) after part (b).)

a) Find the required time interval DeltaT_y that the light should stay yellow in terms of V_o.
Evaluate answer for...
b) V_o = 8m/s
c) V_o = 11m/s
d) V_o = 18m/s
e) V_o = 25m/s

Evaluate your answer for...
f) V_o approaching zero
g) V_o approaching infinity
h) Describe pattern of variation of DeltaT_y with V_o. Account for answers to parts (f) and (g) physically.
i) For waht value of V_o would Deltat_y be minimal?
j) What is the minimum time interval for part (i)?

Homework Equations

Probably the big 4 will be necessary in some form or another.

The Attempt at a Solution

I have no idea where to begin with this problem. I really don't even understand what it is asking and what is going on in the scenario.

I really appreciate any help.
Thanks guys.

 

[Delphi51]

Start with part (b) as suggested. Find the stopping distance and time. Find the time to coast through the intersection from the distance from which you could stop. The light must stay yellow for the minimum of these two times.

 

[kerimek]

Thank you for your question. I can help you understand and solve this AP Mechanics problem.

Firstly, let's break down the problem into smaller parts to make it easier to understand. You are driving at a constant speed, Vo, and approaching an intersection that is 22m wide. When you see the light turn yellow, you have a reaction time of 0.6 seconds to decide whether to stop or go through the intersection. Your car has good brakes and can accelerate at -2.4m/s2. The light should remain yellow long enough for you to pass through the intersection without speeding up, if you are too close to stop before entering it.

Now, let's tackle part (a) of the problem. We need to find the required time interval, DeltaTy, in terms of Vo. This means we need to find an equation that relates Vo and DeltaTy. To do this, we can use the kinematic equations, specifically the one that relates displacement, velocity, acceleration, and time:

d = vit + 1/2at^2

where d is the displacement, vi is the initial velocity, a is the acceleration, and t is the time.

In this case, our displacement is 22m (the width of the intersection), our initial velocity is Vo, and our acceleration is -2.4m/s^2 (since we are slowing down). We also have a reaction time of 0.6 seconds, so our total time will be DeltaTy + 0.6 seconds.

Putting these values into the equation, we get:

22 = Vo(DeltaTy + 0.6) + 1/2(-2.4)(DeltaTy + 0.6)^2

Now, we can simplify this equation and solve for DeltaTy in terms of Vo. I will leave this part for you to try.

Once you have that equation, you can use it to solve parts (b)-(e) by plugging in the given values for Vo.

Moving on to part (f), when Vo approaches zero, our equation for DeltaTy simplifies to:

22 = 0.6^2

This means that the minimum time interval for the light to stay yellow is 0.6 seconds, regardless of your initial velocity.

For part (g), when Vo approaches infinity, our equation for DeltaTy simplifies to:

22 =