Video Transcript
A car is initially at rest. After uniform acceleration of two
meters per second squared, the car has a speed of 19 meters per second. For how much time does the car
accelerate? Give your answer to one decimal
place.
To begin, we should recall the
formula for acceleration. The acceleration of an object is
equal to the change in the object’s speed divided by the time it takes for the
change in speed to occur, which can be written mathematically like this, where 𝑎
represents acceleration. These triangles are Greek symbols
called deltas, which are often used to indicate a change in some quantity.
In order to calculate the
acceleration, we need to know the change in speed, Δ𝑣, and the change in time,
Δ𝑡. But this question is not asking us
to calculate the acceleration of the car. It’s asking us to calculate the
time for which the car accelerates, which is Δ𝑡. So we need to take our equation for
acceleration and make Δ𝑡 the subject. We can do this by rearranging the
equation to get Δ𝑡 by itself on one side.
We’ll start by multiplying both
sides by the change in time, Δ𝑡. We can see that the Δ𝑡-terms in
the numerator and denominator of the right-hand side cancel each other out, leaving
us with the equation Δ𝑡 multiplied by 𝑎 equals Δ𝑣. Next, we need to divide both sides
by the acceleration, 𝑎. This time, on the left, these
𝑎-terms both cancel, leaving us with our final equation for Δ𝑡. We see that the change in time is
equal to the change in speed divided by the acceleration.
So, to calculate the time taken for
the car to accelerate, we need to know the values of the change in the car’s speed
and the car’s acceleration. The question tells us that the car
has a uniform acceleration of two meters per second squared, so this is the value of
𝑎. Next, we need to find the change in
speed. In this question, we are told that
the car is initially at rest. This means the initial speed of the
car is zero. We’re also told that the car’s
final speed is 19 meters per second. The change in speed is equal to the
final speed minus the initial speed. So Δ𝑣 equals 19 meters per second
minus zero meters per second, which is just equal to 19 meters per second. If we substitute these values into
our equation for Δ𝑡, we see that the time for which the car accelerates is equal to
19 meters per second divided by two meters per second squared.
Before we calculate this value,
let’s make sure the units are correct. On the right-hand side of the
equation, we have speed in units of meters per second divided by acceleration in
meters per second squared. We can simplify this by recalling
that dividing by a fraction is equivalent to multiplying by the reciprocal of the
fraction. So the units are equal to meters
per second multiplied by seconds squared per meter. Here, we can cancel both of the
meters terms, leaving us with seconds squared divided by seconds. This is simply equal to seconds,
which is the correct unit for time.
Now, we are ready to solve this
equation. The time it takes for the car to
accelerate is equal to 19 meters per second divided by two meters per second
squared. This comes out to 9.5 seconds, so
we have our answer. To one decimal place, we’ve found
that the car accelerates for a time of 9.5 seconds.
