What is the maximum reaction time of the motorist that will allow her or him to avoid hitting the deer?

Question

A motorist is traveling at 16m/s encounters a deer in the road 41m ahead and the maximum acceleration of the vehicle's brakes are capable of -5m/s^2. Answer in units of s.

Answer 1

First pull out all the information you are given. The initial velocity of the motorist (and presumably his car) is 16 m/s. Putting this into equation form we get;

 

V₀=16 m/s

 

D₀=41 m     Where D₀ is the reported distance between the motorist and the deer when first seen. Hopefully this is really the distance between the front bumper of the car and the deer, or there could be problems.

a=-5 m/s²

The first thing you are going to want to find is the stopping distance for the car. Since you are given the maximum braking acceleration and the initial velocity you can set up an equation for this using the formula below. Remember that when stopped the car will have velocity V equal to 0.

 

V=a*t_m+V₀.

 

Solve the above for t_m and plug in the given values. This is the absolute minimum time in which the car can stop. Hence the little m next to the t in the above formulation. Unfortunately you want the maximum reaction time, so we need to keep going.

 

Recalling that the formula for distance traveled with constant acceleration (and variables replaced to find minimum stopping distance) is;

D_m=1/2*a*(t_m)² + V₀*t_m

 

You will then want to plug in your new t_m value for t and -5 for 'a' to find the minimum stopping distance D_m for the car. Once you have this you will want to find the difference D₀-D_m to determine the distance the car can travel while the motorist is still reacting and still miss the deer. Let's call that D_r for reaction distance. So...

 

D₀-D_m=D_r

 

And finally, to get the requested reaction time into seconds we will use another version of the same distance under constant acceleration formula, and solve for t_r which is finally the requested maximum reaction speed.

 

D_r=(1/2)*a*(t_r)² + V₀*t_r

 

 

I hope this helps. I've left the algebra for you. If you have questions conceptually with what is going on let me know.

Answer 2

Think about the problem backwards. How far does it take to go from 16 m/s to 0 m/s at the acceleration given?
vf^2 = vo^2 + 2 * a * d
Once you find that distance, subtract it from the 41 m. Now you have how many meters you can go before you need to brake. You have a new distance and an initial velocity, which means you can find the time it takes. This is the reaction time. Good luck!