### Video Transcript

A car of mass five metric tons is
moving along a straight horizontal road. The resistance to its motion is
directly proportional to its speed. When the car is traveling at 78
kilometers per hour, the resistance is equal to 40 kilogram-weight per metric ton of
the carโs mass. Given that the maximum force of the
engine is 300 kilogram-weight, determine the carโs maximum speed ๐ฃ and the power ๐
at which its engine operates at this speed.

We will begin by sketching a
diagram to model the situation. We have a car of mass five metric
tons moving along a straight horizontal road. We are told that the resistance to
its motion ๐
is directly proportional to its speed ๐ฃ. Since ๐
is directly proportional
to ๐ฃ, we know that ๐
is equal to some constant ๐พ multiplied by ๐ฃ. And we can calculate this constant
๐พ by dividing the resistance ๐
by the velocity ๐ฃ. We are told that when the car is
traveling at 78 kilometers per hour, the resistance is equal to 40 kilogram-weight
per metric ton. As the car has mass of five metric
tons, the resistance is equal to 200 kilogram-weight as 40 multiplied by five is
200.

We are also told that the maximum
force of the engine is 300 kilogram-weight. This will occur when the car is
traveling at its maximum speed ๐ฃ that we are trying to calculate. As already mentioned, the
resistance here will be equal to ๐พ multiplied by ๐ฃ. As the car is traveling at its
maximum speed, we know that the acceleration will be equal to zero. As well as calculating the maximum
speed ๐ฃ, we need to calculate the power ๐ at which the engine operates at this
speed. We will do this using the formula
๐ is equal to ๐น multiplied by ๐ฃ.

Going back to our first diagram, we
see that the question gives us values in nonstandard units. In order to convert 78 kilometers
per hour into the standard units of meters per second, we recall that there are 1000
meters in one kilometer and 3600 seconds in one hour. This means that we can multiply 78
by 1000 and then divide by 3600. This is the same as 78 divided by
3.6, which is equal to 65 over three or 21.6 recurring meters per second. We also need to convert the
resistance from kilogram-weight to newtons. And one kilogram-weight is equal to
9.8 newtons. Multiplying 200 by 9.8 gives us
1960. The resistance to the carโs motion
is 1960 newtons when its speed is 65 over three meters per second.

We can now use these values to
calculate the constant ๐พ. It is equal to 1960 divided by 65
over three. As dividing by a fraction is the
same as multiplying by its reciprocal, this is the same as 1960 multiplied by three
over 65, which gives us a value of ๐พ equal to 1176 over 13.

Letโs now consider our second
diagram where the car is traveling at its maximum speed. We begin by converting 300
kilogram-weight into newtons. Multiplying 300 by 9.8 gives us
2940. The maximum force of the carโs
engine is 2940 newtons. We can now use this information to
calculate the carโs maximum speed ๐ฃ. Newtonโs second law states that ๐น
equals ๐๐. The sum of the vector forces is
equal to the mass multiplied by the acceleration. There are 1000 kilograms in a
ton. Therefore, the mass of the car is
5000 kilograms. If we take the positive direction
to be the direction of travel, the sum of our forces is 2940 minus 1176 over 13
๐ฃ. This is equal to a mass of 5000
multiplied by an acceleration of zero.

The right-hand side of the equation
is equal to zero. And we can then add 1176 over 13 ๐ฃ
to both sides. Dividing through by 1176 over 13,
we get ๐ฃ is equal to 65 over two or 32.5. The maximum speed of a car is 32.5
meters per second. To convert this back into
kilometers per hour, we can multiply 32.5 by 3.6. This is equal to 117. The maximum speed of the car is 117
kilometers per hour.

We can now calculate the power at
which the engine operates by multiplying the force of 2940 newtons by the velocity
of 32.5 meters per second. This is equal to 95550 watts. Whilst this is the correct value in
standard units, since the speed is in kilometers per hour, we will give the power in
horsepower. We recall that one horsepower is
equal to 735 watts. This means that we need to divide
95550 by 735. This is equal to 130. The power at which the carโs engine
operates at its maximum speed is 130 horsepower. We can therefore conclude that the
two answers to this question are ๐ฃ is equal to 117 kilometers per hour and ๐ is
equal to 130 horsepower.