Video Transcript
A motorcycle of mass 250 kilograms moving at 32 meters per second has four times as much kinetic energy as a 640-kilogram mass car. What is the velocity of the car?
Alright, so, we have a motorcycle, and we know its mass, which we’ll call 𝑚 sub m for motorcycle, is 250 kilograms. We also know that it’s moving at a velocity, which we’ll call 𝑣 sub m, of 32 meters per second. This is a good time to remember that an object in motion, like the motorcycle, has a kinetic energy KE equal to one-half times its mass times its velocity squared. Now, we’re also told that there’s another vehicle with kinetic energy, a car, with a mass, which we’ll call 𝑚 sub c for car, of 640 kilograms. Because it has kinetic energy, the car must be moving, and it’s our job to figure out its velocity.
We can do this by relating the car’s kinetic energy to the motorcycle’s kinetic energy. We’ve been told that the motorcycle has four times as much kinetic energy as the car. Because we already know the mass and velocity of the motorcycle, we could calculate its kinetic energy at any time, but we’ll hold off for now. Since we don’t know the kinetic energy of the car, let’s make this term the subject of the equation by multiplying both sides by one-fourth, which cancels the four out of the right-hand side, so that we have one-fourth times the kinetic energy of the motorcycle equals the kinetic energy of the car.
We also know that we can express the kinetic energy of the car using one-half times mass times velocity squared. Let’s go ahead and make 𝑣 sub c the subject of this formula since that’s what we’re trying to solve for to answer this question. First, we’ll divide both sides by one-half times the mass of the car. This way, the mass of the car cancels out of the right-hand side along with the factor of one-half. We should also note that on the left-hand side the factor of one-half in the denominator can be written as a factor of two in the numerator.
Now, we have two times the car’s kinetic energy divided by the car’s mass equals the car’s velocity squared. Now taking the square root of both sides and then writing the formula so that velocity appears on the left of the equal sign, we have an expression for the velocity of the car written in terms of its mass and kinetic energy. But remember, we don’t explicitly know the kinetic energy of the car, although we do know its value in terms of the kinetic energy of the motorcycle. This we can explicitly calculate. So let’s copy this equation and use the kinetic energy formula to write it in terms of values that we know.
Thus, the kinetic energy of the car equals one-quarter times the kinetic energy of the motorcycle, or one-quarter times one-half times the motorcycle’s mass times the motorcycle’s velocity squared. Now, let’s make this substitution in the formula for the car’s velocity. That way, under the radical, the numerator two times the kinetic energy of the car becomes two times one-quarter times one-half times the motorcycle’s mass times the motorcycle’s velocity squared. We’re getting close now.
Notice that the factors of two and one-half cancel each other out. And because the right-hand side of the formula is written in terms of values that we know expressed in base SI units, we’re ready to substitute them in and calculate. Also, rewriting the one-fourth in the numerator as a four in the denominator, we have that the velocity of the car equals the square root of the mass of the motorcycle times the velocity of the motorcycle squared divided by four times the mass of the car. Looking at the units, we can cancel out kilograms from the numerator and denominator, leaving only the square of meters per second under the radical, or just meters per second, for our final answer.
Finally, grabbing our calculators, this expression comes out to exactly 10 meters per second, and we have our answer. We found that the velocity of the car is 10 meters per second.
