Video Transcript
A car was moving on a straight road
for five hours with an average velocity of 80 kilometers per hour. During the first three hours, the
car was moving with a velocity of 92 kilometers per hour, and for the rest of the
trip, it was moving with a velocity 𝑣. Calculate the velocity 𝑣 of the
car.
In this question, a car is
traveling on a straight road, and we want to find the velocity of the car during the
second part of the trip. Let’s begin by visualizing the
problem. Recall that the velocity 𝑣 of an
object is related to the displacement Δ𝑠 of the object in a time interval Δ𝑡 by
the formula 𝑣 equals Δ𝑠 over Δ𝑡. We will begin by calculating the
distance 𝑑 traveled by the car during the entire trip. Recall that displacement is the
straight-line distance from a point to another point.
Since the car was moving on a
straight road and did not change direction, the displacement of the car is simply
the distance traveled by the car. Similarly, we do not need to
concern ourselves with the direction of the velocity since the car is moving on a
straight road and does not change direction. We can make the displacement Δ𝑠
the subject by multiplying both sides of the equation by Δ𝑡. Doing this, we get Δ𝑠 equals
𝑣Δ𝑡. We have labeled the distance
traveled by the car as 𝑑, so Δ𝑠 is equal to 𝑑. We are told in the question that
the car has an average velocity of 80 kilometers per hour. So 𝑣 is equal to 80 kilometers per
hour. We are also told that the car was
moving for five hours at this average velocity. So Δ𝑡 is equal to five hours.
Substituting these values into the
equation, we find that the distance traveled by the car 𝑑 is equal to 80 kilometers
per hour multiplied by five hours. We need to be extra careful here
since we are not using SI units in this equation. The velocity we are using has units
of kilometers per hour, while the time has units of hours. By multiplying these units
together, we find that the distance will have units of kilometers. So completing the calculation, we
find that the total distance traveled by the car 𝑑 is equal to 400 kilometers.
Now that we know the distance the
car has traveled for the entire trip, we can calculate the distance the car traveled
during the first three hours. Let’s label the time of three hours
as 𝑡 one and the distance traveled during the first three hours as 𝑑 one. We are told in the question that
during the first three hours, the car was moving with a velocity of 92 kilometers
per hour. So the velocity during the first
part of the journey, which we will label as 𝑣 one, is equal to 92 kilometers per
hour. And the time interval here is three
hours.
So by substituting these values
into the equation Δ𝑠 equals 𝑣Δ𝑡, we find that the distance traveled during the
first three hours 𝑑 one is equal to 92 kilometers per hour multiplied by three
hours. Being mindful of units, this
calculation gives 𝑑 one equals 276 kilometers.
Now that we know the total distance
traveled by the car and the distance traveled when the car was moving with a
velocity of 92 kilometers per hour, we can determine the distance traveled by the
car when it was moving with a velocity 𝑣. This distance, which we will label
as 𝑑 two, is simply equal to 𝑑 minus 𝑑 one. Substituting the values for 𝑑 and
𝑑 one into this equation, we find that the distance 𝑑 two is equal to 400
kilometers minus 276 kilometers, which equals 124 kilometers. The car moved for a total of five
hours. The car moved with a velocity of 92
kilometers per hour for the first three hours and with a velocity 𝑣 for the rest of
the trip. So the time interval that the car
traveled with a velocity 𝑣 is equal to five hours minus three hours, which equals
two hours.
With this information, we can use
the original equation 𝑣 equals Δ𝑠 over Δ𝑡, where Δ𝑠 equals 𝑑 two equals 124
kilometers and Δ𝑡 equals two hours, to find the velocity 𝑣. Performing this calculation, we
find that the velocity 𝑣 is equal to 124 kilometers divided by two hours. Being careful with units again, the
distance was given in kilometers. And the time interval was given in
hours. So our velocity will have units of
kilometers per hour. With this in mind, completing the
calculation, we find that the velocity 𝑣 is equal to 62 kilometers per hour. So, this is our final answer to
this question. The velocity 𝑣 of the car is equal
to 62 kilometers per hour.
