Question Video: Relative Speed of Bodies Moving in the Same Direction Mathematics

Two cars 𝐴 and 𝐵 are moving with velocities 41 km/h and 32 km/h, respectively, in the same direction. Determine 𝑣_(𝐴𝐵).

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Video Transcript

Two cars 𝐴 and 𝐵 are moving with velocities 41 kilometers per hour and 32 kilometers per hour, respectively, in the same direction. Determine 𝑣 sub 𝐴𝐵.

Okay, so let’s say that these are our two cars 𝐴 and 𝐵. And we’re told that as these cars move in the same direction, car 𝐴 has a velocity of 41 kilometers an hour and 𝐵, 32 kilometers an hour. Since we’re talking here about velocities, which are vector quantities and can be positive or negative, the fact that both of these velocities are given as positive values tells us that we can consider the direction in which these cars move to be the positive one.

All that being said, we want to calculate 𝑣 sub 𝐴𝐵. Here, the order in which these subscripts appear is very important. Written this way, we want to calculate the velocity of car 𝐴 with respect to car 𝐵. That’s what’s implied by putting the 𝐴 and 𝐵 in this order, so the velocity of car 𝐴 with respect to car 𝐵. To picture this, we can imagine being passengers in car 𝐵 and observing car 𝐴 as it moves relative to us.

The way to go about actually calculating 𝑣 sub 𝐴𝐵 is to take the velocity of the first subscript, the velocity of car 𝐴, and subtract from it the velocity of the second subscript, car 𝐵. 𝑣 sub 𝐴𝐵 equals 𝑣 sub 𝐴 minus 𝑣 sub 𝐵. Just as a side note, if we wanted to compute the opposite, the velocity of car 𝐵, with respect to car 𝐴, then once again we would put the velocity of our first subscript, car 𝐵, first and the velocity of our second subscript second.

Anyway, as we know, it’s 𝑣 sub 𝐴𝐵 that we want to calculate. Leaving off the units, this equals 41 minus 32 or nine. Relative to an observer in car 𝐵 then, car 𝐴 is traveling at nine kilometers an hour. Note also that this is a velocity, so it means car 𝐴 is moving ahead of car 𝐵 at a rate of nine kilometers every hour.