Video Transcript
There are two identical vehicles on
two different roads under the same conditions. The drivers of both vehicles
encounter an obstacle, suddenly brake, and then come to a complete stop. Driver one has a reaction time that
is double the reaction time of driver two. How, if at all, will the braking
distances of the two vehicles be different? (A) Driver one will have half the
braking distance of driver two because braking distance is proportional to reaction
time. (B) Driver one will have twice the
braking distance of driver two because braking distance is inversely proportional to
reaction time. (C) They will be the same because
braking distance doesn’t depend on reaction time.
In this example, we have two
identical vehicles on roads that have the same conditions. These vehicles move along. But suddenly an obstacle, in this
case a crash-landed unidentified flying object, appears in their path. As quickly as possible, the drivers
of these two vehicles will jam on the brakes and bring the vehicles to rest. We note though that it takes some
time for the drivers to respond to the obstacle in front of them and press down on
the brake pedal. The distance each vehicle travels
during that time is called the vehicle’s thinking distance. Here, we’ve drawn these thinking
distances differently because we’re told that the reaction time of the driver in
vehicle one is twice as long as the reaction time of the driver in vehicle two.
Regardless of differences in
thinking distance though, once a vehicle reaches the end of that distance, that’s
the moment that the brake pedal in the car is pressed down. This is where each vehicle begins
its braking distance, the distance it travels while it is coming to a rest. We’re told that these two vehicles
are identical. And this means that they both
decelerate at the same rate so that the braking distance of vehicle one will be the
same as that of vehicle two. The braking distance then doesn’t
depend on either driver. It has to do with the mechanics of
the car. Because the vehicles are the same,
the braking distances will be the same too. We choose answer option (C). The braking distances of these two
vehicles will be the same because braking distance doesn’t depend on reaction
time.
Let’s look now at part two of our
question.
How, if at all, will the thinking
distances of the two vehicles be different? (A) Driver one will have half the
thinking distance of driver two because thinking distance is inversely proportional
to reaction time. (B) Driver one will have twice the
thinking distance of driver two because thinking distance is proportional to
reaction time. (C) They will be the same because
thinking distance doesn’t depend on reaction time.
As we’ve seen, thinking distance
describes the distances our respective vehicles travel while their drivers are
processing the information about the obstacle ahead and preparing to press down on
the brake pedal. Over this distance, our vehicles
maintain a constant speed. This is the speed each vehicle had
before its driver saw the obstacle up ahead, and these speeds are the same.
Let’s recall that, in general, the
speed of an object 𝑣 equals the distance that object travels divided by the time it
takes the object to travel that distance. In our case, the distance is the
thinking distance. And if we multiply both sides of
this equation by the time 𝑡 so that that time cancels out on the right, we see that
distance is equal to speed times time. Or in our scenario, 𝑑 sub t, the
thinking distance of a car, is equal to 𝑣, the speed of that car, times 𝑡 sub r,
the reaction time of the driver.
We’ve said that for each vehicle,
the speed 𝑣 is the same, but we know that the reaction time of the drivers is not
the same. The reaction time of driver one,
we’ll call it 𝑡 sub r one, is twice the reaction time of driver two. We’ll call that 𝑡 sub r two. If we were to calculate the
thinking distance of vehicle one then, we’ll call it 𝑑 sub t one, we can write that
as the speed of the vehicle times two times the reaction time of driver two. We’ve written this equation in
terms of driver two’s reaction time so that we can more easily compare it to the
thinking distance of vehicle two. That distance, we’ll refer to it as
𝑑 sub t two, is equal to 𝑣 times 𝑡 sub r two, the reaction time of driver
two.
We see then that 𝑑 sub t one and
𝑑 sub t two are different from one another by a factor of exactly two. That is, the thinking distance of
vehicle one or the thinking distance of driver one is twice as big as the thinking
distance of driver two or vehicle two. This corresponds to answer choice
(B). Driver one will have twice the
thinking distance of driver two because thinking distance is proportional to
reaction time.
