Video Transcript
In this video, we’re talking about
reaction time. Whenever we try to do something as
quickly as possible, reaction time plays a role. We can define this as the minimum
amount of time required for us to receive some stimulus and take action in
response. Reaction time varies from person to
person, and there’s even a simple experiment we can conduct to measure our own
reaction time.
The way the experiment works is we
have a friend hold a ruler vertically. And then, we place our dominant
hand, whether that’s our right hand or our left hand, at the zero marking on the
ruler. Specifically, we hold out our thumb
and our forefinger around the ruler at this point. So then, if we were looking down on
the ruler, then our thumb and forefinger would fit around it like we see here. Once everything is all set, once
our hand is in position and the ruler is in a stable vertical alignment, then our
friend releases the ruler. And under the influence of gravity,
it begins to drop.
Now, as soon as our eye — let’s say
our eye is over here watching the ruler — sees that the ruler is beginning to
descend, our brain sends a signal to our thumb and forefinger to catch the ruler as
soon as possible. Once that signal arrives, our thumb
and index finger clamp down and we stop the ruler in its descent. And because we’re catching a ruler,
we’re able to see at what vertical position we catch it. And since we know that our thumb
and forefinger started at a position of zero on the ruler, we can solve for the
vertical distance that the ruler fell before we stopped it.
Now, once we know that distance 𝑑,
we can use it in combination with a kinematics equation, also called a SUVAT
equation, to solve for the time that the ruler must have been falling in order to
fall this distance 𝑑. And to a good approximation, that
time is our reaction time. That’s how long it took us to react
to our observation that the ruler was falling.
Now, in order for this to be as
accurate a determination of our reaction time as possible, there are a few factors
we need to keep in mind. One is that the ruler should be
kept as vertical as possible all throughout the experiment. If the ruler was slightly tilted
off of the vertical, that would make our measurement of the distance 𝑑 and,
therefore, our reaction time 𝑡 less accurate.
Along with this, the mass of the
ruler and the material it’s made of may have an impact. Even though rulers with different
masses would fall at the same rate, a heavier ruler made of a relatively smooth
material would be harder to grab and stop quickly. This would have the effect of
increasing our measured distance 𝑑 and, therefore, the reaction time 𝑡 as
well.
Along with all this, it’s important
to use a ruler whose markings are spaced an appropriate distance apart for this
experiment. For example, a ruler with markings
spaced far apart like this one would make it difficult to precisely measure 𝑑 and,
therefore, get a precise result for the reaction time 𝑡.
This ruler-drop experiment is one
way to measure our reaction time. And now, let’s consider a fairly
common scenario where reaction time impacts object motion. The situation we’re thinking of is
a car out driving on the road. Let’s say this car is moving along
at a constant speed to the right. We’ll call that speed 𝑠. Say that as the car is moving at
this constant speed, the driver suddenly becomes aware of an obstacle farther on
down the road. Perhaps this obstacle is a large
rock that fell in the road, and it’s posing a safety hazard to any traffic on the
road.
To avoid this hazard, the driver
jams on the brakes and stops the car as quickly as possible. We can see how this process of
bringing the car to a stop involves the driver’s reaction time. But in contrast to our ruler-drop
experiment, in this case, the driver’s reaction time isn’t the only thing we need to
take into account to figure out the total distance this car will travel before it
comes to a stop. Here’s how we can think of this
process.
At some time value — we’ll call it
𝑡 sub zero — the driver becomes aware, for the first time, of this obstacle. We could say that, at 𝑡 equals 𝑡
sub zero, light from the obstacle first reaches the driver’s eyes. At some later time — we can call it
𝑡 sub brake — the driver will begin for the first time to push down on the brake of
the car. Now, the amount of time that
separates 𝑡 sub zero from 𝑡 sub brake is the driver’s reaction time. And all during this time interval,
all during the driver’s reaction time, before the brake is pressed, the car is
continuing to move down the road at this constant speed.
So, by the time the driver reacts
and begins to press down on the brake, the car will be some distance further down
the road. And it will still be moving at this
speed 𝑠. This distance, here, that the car
has traveled during this time has a special name. It’s called the thinking distance
of the car. This is the distance that the car
travels while the driver is reacting, in other words, responding to this information
that there’s an obstacle ahead and getting to the point of pressing down on the
brake.
Since the car moves along at a
constant speed all while the driver is reacting, we can see that the longer the
reaction time is, the greater the thinking distance traveled by the car will be. And that’s because all throughout
the driver’s reaction time, the car continues to move at a steady speed down the
road. But as we’ve said, eventually the
driver presses down on the brake as hard as possible to bring the car to a stop as
soon as possible. Once the brake is pressed, we can
no longer say the car is moving at the speed 𝑠.
We know that, instead, the car is
slowing down to a stop. That stopping though doesn’t happen
instantaneously. So, if we call the moment in time
when the car comes to a full and complete stop 𝑡 sub stop, then just like before,
there’s a time interval between this time and the last one we mentioned, 𝑡 sub
brake. Throughout this period, the car is
slowing to a stop. But it’s still moving forward on
the road. And that means it covers some
distance. The specific name for it is the
braking distance of the car. That is, it’s the distance the car
travels while its brakes are on.
So, if we think about the total
distance that this car travels before it comes to a stop. We can refer to that as the
stopping distance. Then, we can see that the stopping
distance is equal to the sum of the thinking distance and the braking distance. Now, let’s think about how these
two different distances are affected or not by the reaction time of our driver.
If our driver’s reaction time is
very slow — that is, this time interval is relatively long — then that means the
distance the car travels before the brake is pressed will be relatively long
too. That is, as reaction time goes up,
thinking distance increases too. And in fact, if we remember the
relationship that a constant speed 𝑠 is equal to a distance traveled divided by the
time it takes to travel that distance. Then, multiplying both sides of
this equation by the time 𝑡 so that it cancels out on the right. We can see that, for a constant
speed 𝑠 like our car is moving at before the brakes are pressed, the time 𝑡 if we
multiply it by some constant value — in this case, the speed — is equal to the
distance 𝑑.
Now, it’s important to realize that
this equation is only true so long as 𝑠 is constant. Related to our scenario, it’s only
the case over the thinking distance of the car. Once the driver presses the brake,
𝑠 is no longer constant, and so this equation no longer holds. But while it does hold, while it is
true, we can think of this time 𝑡 as the reaction time of the driver. And then, we can think of this
distance 𝑑 as the thinking distance of the car.
So, if we take the driver’s
reaction time and we multiply it by a constant value, then that’s equal to the
thinking distance traveled by the car. And then, mathematically, we can
write this equation another equivalent way. We can say that the reaction time
𝑡 is directly proportional to the thinking distance 𝑑. Which means that if we double the
reaction time, say, then we’ll double the thinking distance. Or if we cut the driver’s reaction
time in half, then we’ll divide the thinking distance by two, and so on. By whatever factor the reaction
time changes, the thinking distance changes by that same factor.
So, we can see that there’s this
relationship between the driver’s reaction time and the thinking distance traveled
by the car. These two values are directly
proportional to one another. But this may make us wonder, what
about reaction time and the braking distance of the car? As it turns out though, these two
values are independent of one another.
We can see that by considering that
the braking distance of the car corresponds to this time interval, between 𝑡 sub
brake and 𝑡 sub stop. And that that interval of time
doesn’t have to do with the driver’s reaction time. The factors that affect the car’s
braking distance are things like how fast the car is initially moving, how effective
the braking system of the car is, the condition of the road and of the car’s tires,
and factors along those lines. The braking distance of the car is
specifically not affected by the driver’s reaction time.
Now that we’ve seen how the
thinking distance and braking distance of the car are related to the driver’s
reaction time, let’s consider how these two distances are related to the car’s
initial speed 𝑠. Going back over to our equation for
time, speed, and distance, we can see right away that the car’s thinking distance,
this value 𝑑, is directly proportional to the car’s initial speed 𝑠.
We can see this by letting the
driver’s reaction time 𝑡 stay constant. And then, under that condition, say
that we doubled the initial speed of the car 𝑠. Since 𝑡 is the same and 𝑠 has
gotten twice as big, for this equation to hold, that must mean that 𝑑, the thinking
distance, would double as well. Or similarly, if 𝑠 went up by a
factor of four, then again since reaction time is the same, 𝑑 would have to go up
by a factor of four too. We can express this in words like
this. We can say that thinking distance
is proportional to initial speed.
And then, going further, we wonder
how braking distance depends on initial car speed. Thinking about this, if we assume
that our car is decelerating constantly over the braking distance. That is, it has a constant
deceleration all throughout this time interval. Then, that means we can describe
its motion over the braking distance using a kinematic equation of motion.
If we call the braking distance of
the car 𝑑 sub 𝑏, then this equation of motion says that the braking distance of
the car is directly proportional to the initial speed of the car squared. This means that, for example, if we
were to double the car’s initial speed, then we would quadruple its braking
distance. Or if the initial speed of the car
was four times higher, then the braking distance would be 16 times greater, four
squared.
We can write out that relationship
like this. We can say that braking distance is
proportional to initial speed squared. So then, in this scenario of a car
moving along at a constant speed and then coming to a stop as soon as the driver is
able to stop the car. We can see how these two distances,
the thinking distance and braking distance of the car, relate to the driver’s
reaction time as well as the initial speed of the car. Knowing all this, let’s look now at
an example exercise about reaction time.
The thinking distance and
braking distance for a car at different initial speeds are shown by the lengths
of the two-colored bars in the diagram. The longer the bar, the greater
the initial speed the car stops from. Which of the following
quantities is shown by the length of the gray part of the bar? a) Stopping distance. b) Thinking distance. c) Braking distance.
All right, here, in our
diagram, we see these two-colored bars, purple and gray. We can see that as we go from
top to bottom, the length of the bars increases. And we’re told that the longer
the bar, the greater the initial speed of the stopping car that the bar refers
to. So, let’s say that this here is
our car and that, at the outset, it’s moving at some speed — we can call it 𝑠 —
down the road. And let’s say that, at this
moment in time, the driver of the car sees a reason to bring the car to a
stop.
However, while the driver is
reacting to this information and getting ready to press down on the brake, the
car continues to travel on the road moving at its initial speed 𝑠. During this time, the car
covers a distance called the thinking distance. It’s the distance the car
travels while the driver is thinking about pressing the brake pedal. Then, at the end of the
reaction time, the driver does press the brake pedal down, and the car begins to
stop.
But while the car is slowing
down, it will still continue to travel some amount of distance called the
braking distance. And then, at the end of the
braking distance, the car has come to a stop. Its velocity is zero. It’s these two distances, the
thinking and braking distances, that are shown in this diagram. And we want to figure out
whether the length of the gray part of the bars in the diagram refers to the
stopping distance of the car, the thinking distance, or the braking
distance.
Now, this first option, option
a) stopping distance, is a term that refers to the sum of the thinking and the
braking distances. On our sketch, this total
distance here that the car takes to come to a stop is called the stopping
distance. But because we’re told that
it’s the thinking distance and the braking distance that are shown in the
lengths of the two-colored bars in the diagram. We know that it can’t be the
case that the length of the gray part of this bar refers to anything other than
one of those two distances. Stopping distance is indicated
by the total length of each bar, the gray part and the purple part. So, we can cross out answer
option a).
Now, going back to our diagram,
if we consider a vertical line between these two points, then we can see that
both the gray part of the bars to the right and the purple part of the bars to
the left move out from this vertical line. And as we noticed earlier, the
longer these lines are from what we could call their beginning point here on
this vertical line, the greater the initial speed of the car that’s coming to a
stop. This makes sense because if we
think back to the speed of our stopping car 𝑠, if that speed was very high,
then it makes sense that the thinking distance will be longer and that the
braking distance will be longer too.
So, on our diagram, as we go
from the top to the bottom, our car’s initial speed is increasing. And therefore, the thinking and
braking distances are increasing too. Now, as we specifically
consider the length of the gray part of the bar over on the right of our
vertical line, we want to know whether those lengths refer to the thinking
distance or the braking distance of the car. To figure that out, let’s
consider how the lengths of these gray bars change as we go from the top to the
bottom of the diagram.
Comparing these line lengths
with one another, we can see that the difference in the lengths is roughly
constant as we go from top to bottom. On the other hand, on the other
side of our diagram, with the purple parts of the bars, those length differences
seem to increase as we go from the top to the bottom of our diagram. So, that’s an important
difference between the gray and the purple parts of our bars. That as the car’s initial speed
increases, the length of the purple parts of the bars increases at a greater
rate than the length of the gray parts.
And this brings us back to
thinking distance and braking distance. We can recall that there’re
specific mathematical relationships between these two distances and the car’s
initial speed, what we’ve called 𝑠. If we call the car’s thinking
distance 𝑑 sub 𝑡, then we can remember that this is directly proportional to
the car’s initial speed 𝑠. That is, as one of these
variables changes by a certain factor, the other changes by the same factor.
But then, if we consider the
braking distance of the car — we can call it 𝑑 sub 𝑏 — that distance is
proportional not to 𝑠 but to 𝑠 squared. This means, for example, that
if we were to double the initial speed of our car, then we will quadruple the
braking distance. So, returning to our diagram,
we can see that the purple parts of these bars increase in length at a greater
rate, we could say, than the gray parts of the bars. And looking at these equations,
we can see that 𝑑 sub 𝑏 would increase at a greater rate with increasing
initial speed 𝑠 than 𝑑 sub 𝑡, the thinking distance.
This tells us that because the
length of the gray parts of these bars increases at a slower rate as the speed
of the car goes up and up, then this part of the bar must refer to the thinking
distance of the car rather than the braking distance. That’s because, as we saw, the
thinking distance is proportional to the car’s initial speed, whereas the
braking distance is proportional to its initial speed squared. And now, we’re able to answer
our question. The length of the gray part of
the bar in our diagram refers to the thinking distance of the car.
Let’s summarize now what we’ve
learned about reaction time. Starting off, in this lesson, we
learned that reaction time indicates the quickness of human response to an incoming
sensory signal. We also learned in studying the
ruler-drop experiment that reaction time can be measured using a carefully dropped
ruler. And we also learned about a
scenario where a car is traveling at a constant speed and then comes to a stop.
In this case, the total distance
the car takes to come to a stop — called it stopping distance — is equal to the sum
of the thinking distance of the car plus its braking distance. The thinking distance of a car is
how far the car moves during the driver’s reaction time. So that the longer the reaction
time, the longer the thinking distance and, therefore, the longer the stopping
distance of the car.
And in addition, we saw that if 𝑑
sub 𝑡 refers to the thinking distance of the car, then this distance is
proportional to the driver’s reaction time. While if 𝑑 sub 𝑏 refers to the
braking distance of the car, then this distance is independent of reaction time. This is a summary of reaction
time.
