### Video Transcript

The driver of a car travelling at
30 metres per second has a reaction time of 1.5 seconds. The car’s brakes decelerate the car
at 3.75 metres per second squared once they are activated. How much time does the car take to
stop, including the driver’s thinking time?

Okay, so, in this question, we’ve
been told that we’re dealing with a car. So, let’s say this block is going
to represent our car. And we’ve been told that it’s
travelling at 30 metres per second initially. So, let’s assume it’s initially
travelling towards the right at 30 metres per second.

Now we’ve been told that the driver
has a reaction time of 1.5 seconds. This means that if the driver
notices a hazard at this point, then the car continues to move for another 1.5
seconds before the driver can actually press the brakes. And this is because the driver’s
reaction time is 1.5 seconds. And therefore, it took them that
long to react to the hazard that they noticed 1.5 seconds earlier. So, until the brakes are pressed,
the car is still travelling towards the right at 30 metres per second. And we can say that the time
interval here is 1.5 seconds as we’ve been told in the question

Now after the driver’s pressed the
brakes, we know that the car will decelerate. And at some point in the future,
the car will eventually stop. So, we can say that at this
position the car speed is zero metres per second. And as well as this, we know that
once the driver presses the brakes, the car will decelerate at 3.75 metres per
second squared. In other words, we can say that
just after the brakes are pressed, the car’s acceleration becomes negative 3.75
metres per second squared. And the reason that it’s negative
is because the car is decelerating. In other words, it’s slowing
down.

Now what we’ve been asked to do is
to find the amount of time taken for the car to stop. And this is including the driver’s
thinking time. Now the driver’s thinking time is
simply the reaction time of 1.5 seconds. So, we need to find the total time
taken from when the driver notices the hazard to when the car has stopped. And so, in order to do that,
because we already know that this amount of time is 1.5 seconds, we need to go about
finding this time interval.

So, let’s say that the time taken
between when the brakes are pressed and when the car actually stops is the time
interval 𝑡. And during this entire time
interval, the car has an acceleration of negative 3.75 metres per second squared
because, remember, this time interval starts when the brakes are pressed. And as well as this, we know the
initial velocity of the car, which is 30 metres per second, and the final velocity
of the car, which is zero metres per second.

So, based on all the information we
know, we need to find an equation that links all of that information in order to
find the time 𝑡. And the equation that we’re looking
for is the definition of acceleration.

We can recall that acceleration is
defined as the change in velocity of an object divided by the time taken for that
change in velocity to occur. Now in this case, we know the
acceleration of the car, which is 𝑎, over the period of time, which is 𝑡. And as well as this, we know the
initial and final velocities. So, we can use this information to
calculate the value of 𝑡.

We can firstly recall that the
change in velocity is found by subtracting the initial velocity from the final
velocity. And now what we can do is to
rearrange this equation to solve for 𝑡. To do this, we multiply both sides
of the equation by 𝑡 over 𝑎. This way, on the left-hand side the
acceleration cancels, and on the right-hand side the time 𝑡 cancels.

And what we’re left with is that
the time 𝑡 is equal to the final velocity minus the initial velocity divided by the
acceleration 𝑎. Then we can substitute in all our
values. The final velocity of the car is
zero metres per second. The initial velocity of the car is
30 metres per second. And the acceleration is negative
3.75 metres per second squared.

And now we can see that because
we’ve taken care to put a negative sign in front of the acceleration, that our value
for time is going to be positive because we’ve got zero minus a positive value. And so, the numerator is going to
be negative and the denominator is already negative. Now a negative value divided by a
negative value gives a positive value. And when we evaluate the right-hand
side, we find that this time interval 𝑡 is eight seconds.

So, we found the time taken between
when the brakes are pressed and when the car actually stops. That time interval is eight
seconds. And now since we’ve been asked to
find the time taken for the car to stop including the driver’s thinking time, all we
need to do is to add up this time interval to this time interval.

And so, we can say that the total
time taken for the car to stop, which we’ll call 𝑡 subscript tot, is equal to the
first time interval, which is 1.5 seconds, plus the second time interval, which is
eight seconds, which is the amount of time the car takes to stop once the brakes
have been pressed. And once we evaluate the right-hand
side of the equation, we’ve found the answer to our question. The time taken for the car to stop,
including the driver’s thinking time, is 9.5 seconds.