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
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
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.