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Question Video: Calculating Power Based on Force and Speed Mathematics

A car of mass 5 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 km/h, the resistance is equal to 40 kg-wt per metric ton of the car’s mass. Given that the maximum force of the engine is 300 kg-wt, determine the car’s maximum speed 𝑣 and the power 𝑃 at which its engine operates at this speed.

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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 car’s mass. Given that the maximum force of the engine is 300 kilogram-weight, determine the car’s maximum speed 𝑣 and the power 𝑃 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 𝑅 is directly proportional to its speed 𝑣. Since 𝑅 is directly proportional to 𝑣, we know that 𝑅 is equal to some constant 𝐾 multiplied by 𝑣. And we can calculate this constant 𝐾 by dividing the resistance 𝑅 by the velocity 𝑣. 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 𝑣 that we are trying to calculate. As already mentioned, the resistance here will be equal to 𝐾 multiplied by 𝑣. 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 𝑣, we need to calculate the power 𝑃 at which the engine operates at this speed. We will do this using the formula 𝑃 is equal to 𝐹 multiplied by 𝑣.

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 car’s motion is 1960 newtons when its speed is 65 over three meters per second.

We can now use these values to calculate the constant 𝐾. 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 𝐾 equal to 1176 over 13.

Let’s 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 car’s engine is 2940 newtons. We can now use this information to calculate the car’s maximum speed 𝑣. Newton’s second law states that 𝐹 equals π‘šπ‘Ž. 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 𝑣. 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 𝑣 to both sides. Dividing through by 1176 over 13, we get 𝑣 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 car’s engine operates at its maximum speed is 130 horsepower. We can therefore conclude that the two answers to this question are 𝑣 is equal to 117 kilometers per hour and 𝑃 is equal to 130 horsepower.

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