A motorist traveling at 16 m/s encounters a
deer in the road 43 m ahead.
If the maximum acceleration the vehicle’s
brakes are capable of is −6 m/s2, what is the
maximum reaction time of the motorist that
will allow her or him to avoid hitting the deer?
To find the maximum reaction time of the motorist, we need to consider the distance traveled by the vehicle while the motorist reacts to the situation.
Let's assume that the motorist takes time 't' to react, during which the vehicle continues to move at its initial velocity of 16 m/s.
The distance traveled by the vehicle during the reaction time 't' can be calculated using the formula:
Distance = Initial Velocity × Time
So the distance traveled by the vehicle during the reaction time 't' is:
Distance = 16 m/s × t
The total distance covered by the vehicle, including the distance traveled during the reaction time, should be equal to or less than the distance to the deer (43 m).
Therefore, we can express this as an equation:
Distance traveled during reaction time + Distance traveled after reaction time = Distance to deer
16 m/s × t + (1/2) × acceleration × t^2 = 43 m
Since the maximum acceleration the vehicle's brakes are capable of is -6 m/s^2 (negative because it is deceleration in this case), we can substitute the value of acceleration as -6 m/s^2 and solve the equation for 't'.
16t - 3t^2 = 43
Now, we have a quadratic equation. Let's solve it to find the values of 't'.
Rearranging the equation:
3t^2 - 16t + 43 = 0
Using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
Substituting the values:
t = [-(16) ± √((16)^2 - 4(3)(43))] / (2(3))
t = [-(16) ± √(256 - 516)] / 6
t = [-(16) ± √(-260)] / 6
As the value inside the square root is negative, there are no real solutions to this equation. It means there is no positive reaction time 't' that will allow the motorist to avoid hitting the deer.
Therefore, the maximum reaction time that will allow the motorist to avoid hitting the deer is zero.