Question Video: Quantities and Units in Mechanics Mathematics

Video Transcript

Velocity is equal to the distance covered over a period of time. Which of the following is not a unit of velocity? Is it (a) centimeters per second, (b) meters per second, (c) meters per second squared, (d) meters per minute, or (e) kilometers per hour?

Since velocity is equal to the distance covered over a period of time, velocity 𝑣 is given by the equation 𝑣 equals 𝑑 over 𝑡, where 𝑑 is the distance and 𝑡 is the time elapsed. We can determine the dimension of velocity by performing dimensional analysis on this equation. All this means is replacing every variable on the right-hand side of the equation with its dimension. So, in this case, the distance 𝑑 has dimension length 𝐿 and the time elapsed 𝑡 of course has dimension time 𝑇. Both sides of the equation must have the same dimension. Therefore, the dimension of 𝑣 must be the same as the dimension of 𝑑 over 𝑡, 𝐿 over 𝑇.

We can now compare this with every one of our answers to see which one does not have units representing dimension L over 𝑇. So, for answer (a), we have the centimeter, which is a multiple of the meter, the base unit for length. And we have second, which is the base unit for time. So answer (a) is 𝐿 over 𝑇. So this is indeed a unit for velocity.

For answer (b), we just have the base unit for length over the unit for time. So this is also a unit for velocity. For answer (c), we have the base unit for length, the meter, divided by the base unit for second squared. So the dimension is 𝐿 over 𝑇 squared. Therefore, (c) is not a unit for velocity. For completeness, for answer (d), we have the base unit for length, again, divided by the minute, which is a multiple of the second. So we have length over time once again. And likewise, for (e), we have the kilometer, a multiple of the meter, divided by the hour, a multiple of the second. So we once again have 𝐿 over 𝑇. Therefore, our answer is (c). Meters per second squared is not a unit of velocity.