Question Video: Quantities and Units in Mechanics Mathematics • Second Year of Secondary School

Video Transcript

The kinetic energy, which is measured in joules, is given by the rule 𝑇 equals one-half 𝑚𝑣 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: 𝑇 equals one-half 𝑚𝑣 squared, where 𝑇 is the kinetic energy, 𝑚 is the mass, and 𝑣 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 𝑇 must be equal to the dimension of one-half times the dimension of 𝑚 times the dimension of 𝑣 squared. One-half is just a number. Therefore, it is dimensionless, or dimension one. 𝑚 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 squared’s 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 𝑑 divided by the time elapsed 𝑡. We now have velocity expressed purely in terms of base SI quantities, length and time. The displacement has dimension length 𝐿 and the time elapsed has dimension time 𝑇, not to be confused with the 𝑇 for kinetic energy. Therefore, 𝑣 squared equal to 𝑑 squared over 𝑡 squared has dimension 𝐿 squared over 𝑇 squared. Putting all these together in the original equation for kinetic energy, we get the dimension of kinetic energy 𝑇 is equal to one times 𝑀 times 𝐿 squared over 𝑇 squared.

We can now substitute in the SI base unit for each of these quantities. Dimension one has no specific unit. For mass 𝑀, we have the kilogram. For length 𝐿, we have the meter, in this case squared. And for time 𝑇, 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.