A ranger in a national park is driving along at 17.9 m/s when a deer jumps into the road 65. m ahead of the vehicle. After a reaction time of t seconds, the ranger applies her brakes, which produce an acceleration of -2.8 m/s2. What is the maximum reaction time she can have if she is to just miss hitting the deer?
s1=v²/2•a=19.9²/2•2.8=57.2 m
Δs=65-57.2 =7.78 m
Δt= Δs/v=7.78/17.9=0.43 s
To determine the maximum reaction time, we need to calculate the time it takes for the ranger to reach the deer if she didn't apply the brakes. This is called the "braking time".
First, let's calculate the time it takes for the ranger to travel the 65 meters when she maintains a constant velocity of 17.9 m/s. We can use the formula:
time = distance / velocity
Substituting the values:
time = 65 m / 17.9 m/s
Now, we have the total time it takes for the ranger to reach the deer, including her reaction time. It can be expressed as:
total time = reaction time + braking time
The reaction time is given as "t" seconds. Therefore, the braking time can be determined by subtracting the reaction time from the total time:
braking time = total time - reaction time
Finally, the maximum reaction time occurs when the braking time is just enough to stop the ranger from hitting the deer. In this case, the total time and braking time will be the same. So, we can set up the following equation:
total time - reaction time = braking time
Plugging in the values, we get:
total time - t = braking time
Since the maximum reaction time is when the braking time equals the total time, we can set up this equation:
total time - t = total time
Simplifying the equation, we get:
-t = 0
Since t can't be negative, the only value that satisfies this equation is t = 0. Therefore, the maximum reaction time the ranger can have to just miss hitting the deer is 0 seconds, meaning she needs to respond instantly to avoid the collision.