### Video Transcript

The kinetic energy, which is
measured in joules, is given by the rule π equals one-half ππ£ squared. Which of the following units is
equal to the joule? (a) Kilogram per meter per second
squared. (b) Kilogram times meter per
second. (c) Kilogram times meter squared
per second squared. (d) Kilogram per meter squared per
second squared. Or (e) kilogram times meter per
second squared.

Consider the equation for the
kinetic energy: π equals one-half ππ£ squared, where π is the kinetic energy, π
is the mass, and π£ is the velocity. In an equation representing
physical quantities, the dimensions of the quantities on both sides must be the
same. Therefore, the dimension of the
kinetic energy π must be equal to the dimension of one-half times the dimension of
π times the dimension of π£ squared. One-half is just a number. Therefore, it is dimensionless, or
dimension one. π is just mass, so it has
dimension mass. Velocity squared is a little more
complicated since it is not a base SI dimension.

To find velocity squaredβs
dimension in terms of the SI base dimensions, we will need to consider the equation
for velocity as well. Recall that the velocity is given
by the displacement π divided by the time elapsed π‘. We now have velocity expressed
purely in terms of base SI quantities, length and time. The displacement has dimension
length πΏ and the time elapsed has dimension time π, not to be confused with the π
for kinetic energy. Therefore, π£ squared equal to π
squared over π‘ squared has dimension πΏ squared over π squared. Putting all these together in the
original equation for kinetic energy, we get the dimension of kinetic energy π is
equal to one times π times πΏ squared over π squared.

We can now substitute in the SI
base unit for each of these quantities. Dimension one has no specific
unit. For mass π, we have the
kilogram. For length πΏ, we have the meter, in
this case squared. And for time π, we have the second,
in this case also squared. Putting all these together, we get
the unit for kinetic energy in terms of the SI base units: kilogram meter squared
per second squared. Comparing this with our possible
answers, we can see this matches with (c) kilogram times meter squared per second
squared.