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Video: Comparing the Speeds of Different Mass Objects with Equal Kinetic Energy

Ed Burdette

How fast must a 3 000-kg elephant move to have the same kinetic energy as a 65.0-kg sprinter running at 10.0 m/s?

03:35

Video Transcript

How fast must a 3000-kilogram elephant move to have the same kinetic energy as a 65.0-kilogram sprinter running at 10 point zero meters per second.

In this problem, we’ll assume that the mass of the elephant, 3000 kilograms, is an exact number. In this problem, we’re working with two different objects: the elephant and the sprinter.

Each of these objects has a mass; we’ll call the elephant’s mass 𝑚 sub e and the sprinter’s mass 𝑚 sub s.

And we’re also told that each of these objects is moving a certain speed; the elephants speed, which we’ll call 𝑣 sub e, is what we want to sell for; and the sprinter’s speed we’ll call 𝑣 sub s.

Now the piece of information were given that will help us solve for 𝑣 sub e, the elephant’s speed, is the fact that we’re told that the elephant and the sprinter have the same kinetic energy. Let’s recall the equation for the kinetic energy of an object. The kinetic energy of an object, which we represent as capital K capital E, is equal to half the mass of the object times its velocity squared. As it relates to our problem, we’re told that the kinetic energy of the elephant, which we’ll call KE sub e, is equal to the kinetic energy of the sprinter, which we’ll call KE sub s.

Referring to our equation for kinetic energy, let’s replace KE sub e and KE sub s with the masses and speeds of our two objects. So we arrive at an equation which says half the mass of the elephant multiplied by its velocity squared is equal to half the mass of the sprinter multiplied by the sprinter’s velocity squared.

Now since we’re trying to solve for 𝑣 sub e, the speed of the elephant, let’s rearrange our equation to that end. We can start by canceling out the one-half on either side.

Next we can divide both sides by the mass of the elephant, 𝑚 sub e. Taking this step cancels out the mass of the elephant on the left side of our equation.

So finally if we take the square root of both sides of our equation, then the square root and square term on the left-hand side of equation cancel out, leaving us with 𝑣 sub e by itself

We can write a simplified and cleaner version of this equation as follows. 𝑣 sub e, the speed of the elephant, is equal to the speed of the sprinter times the square root of the ratio of the mass of the sprinter to the mass of the elephant. Now it’s time to plug in the values we know. 𝑣 sub s is 10 point zero meters per second answer; 𝑚 sub s is 65 point zero kilograms; and 𝑚 sub e is 3000 kilograms.

When we plug these numbers into our calculator, we find a speed of the elephant of one point four seven meters per second. That’s the speed the elephant would have to move at for the kinetic energy of the elephant and the kinetic energy of the sprinter to match.