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2.5Addition of Velocities

Addition of velocities to describe relative motion

Since absolute motion cannot be unambiguously measured, the only

way to describe motion unambiguously is to describe the motion of one

object relative to another. Symbolically, we can write v

PQ

for the velocity of

object P relative to object Q.

Velocities measured with respect to different reference points can be

compared by addition. In the figure below, the ball’s velocity relative to the

couch equals the ball’s velocity relative to the truck plus the truck’s velocity

relative to the couch:

v

BC

=

v

BT

+v

TC

= 5 cm/s + 10 cm/s

The same equation can be used for any combination of three objects, just

by substituting the relevant subscripts for B, T, and C. Just remember to

write the equation so that the velocities being added have the same sub-

script twice in a row. In this example, if you read off the subscripts going

from left to right, you get BC...=...BTTC. The fact that the two “inside”

subscripts on the right are the same means that the equation has been set up

correctly. Notice how subscripts on the left look just like the subscripts on

the right, but with the two T’s eliminated.

These two highly competent physicists disagree on absolute velocities, but they would agree on relative

velocities. Purple Dino considers the couch to be at rest, while Green Dino thinks of the truck as being at rest.

They agree, however, that the truck’s velocity relative to the couch is v

TC

=10 cm/s, the ball’s velocity relative

to the truck is v

BT

=5 cm/s, and the ball’s velocity relative to the couch is v

BC

=v

BT

+v

TC

=15 cm/s.

In one second, Green Dino and the

truck both moved forward 10 cm, so their

velocity was 10 cm/s. The ball moved

forward 15 cm, so it had v=15 cm/s.

Purple Dino and the couch both

moved backward 10 cm in 1 s, so they

had a velocity of -10 cm/s. During the same

period of time, the ball got 5 cm closer to

me, so it was going +5 cm/s.

Section 2.5Addition of Velocities