# Weird kinematics question

## Homework Statement

(3) A motorist is approaching a green traffic light with speed $v_0$ when the light turns
to amber.
(a) If his reaction time is $\tau$, during which he makes his decision to stop and applies his foot to the
brake, and if the maximum braking deceleration is a, what is the minimum distance $s_{min}$ from the
intersection at the moment the light turns to amber in which he can bring his car to a stop?
(b) If the amber light remains on for a time t before turning red, what is the maximum distance
$s_{max}$ from the intersection at the moment the light turns to amber such that he can continue into the
intersection at speed $v_0$ without running the red light?
(c) Show that if his initial speed $v_0$ is greater than

$v_{0_{max}} = 2a(t- \tau )$

there will be a range of distance from the intersection such that he can neither stop in time nor
continue through the intersection without running the red light.

## Homework Equations

kinematics equations

## The Attempt at a Solution

A using $vf^2=v_0^2 +2(-a)d$ is $s_{min}=\frac{v_0^2}{2a} +v_0\tau$

B is simply $v_0 t$

C i'm almost clueless. i can sub the given expression into both of my derived equations for $s_{min}$ and $s_{max}$ and all i get is that they both equal $2at(t-\tau)$ which i can't see how to use to prove that $s_{max}<s<s_{min}$.

This is actually an intermediate mechanics question but seems simple enough.

Substitute your values into the inequality for smax & smin.
Solve for v.

the implication needs to go the other way, if p is the statement about velocity and q is the statement about distance i need to prove p -> q, that would prove q->p

even if i did do as you said, how would i "solve" for v?

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(c) Show that if his initial speed $v_0$ is greater than

$v_{0_{max}} = 2a(t- \tau )$

there will be a range of distance from the intersection such that he can neither stop in time nor
continue through the intersection without running the red light.

s > s_max or s < s_min
v*t > s_max or v*t < s_min

Since we are looking for the maximum initial speed, then
vo_max*t = vo_max^2/(2a) + vo_max*r
t = vo_max/(2a) + r
t - r = vo_max/(2a)
vo_max = 2a(t-r)

i don't understand what you've done

why did you equate the two?