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Relative Velocity

Kinematics

How fast is that car going? The answer depends on where you are standing. From the roadside it might be doing 30 m/s; from another car travelling alongside it, it might seem almost still. Relative velocity is the velocity of one object as measured from another moving object, and it explains everything from overtaking on a motorway to crossing a river in a boat.

What relative velocity is

Relative velocity is the velocity of an object measured by an observer who is also moving. If you and the object move in the same direction, the relative velocity is the difference of your speeds; if you move toward each other, it is the sum. In one dimension, the velocity of A relative to B is **v_AB = v_A − v_B**. The idea is the same in two dimensions, but the subtraction is done with vectors, so direction matters. This is why a car overtaking you slowly seems to crawl past even though both of you are moving fast relative to the road.

Adding and subtracting velocities

To find a relative velocity, choose a frame of reference and subtract the observer's velocity from the object's velocity. For objects moving along the same line, take one direction as positive and keep the signs consistent. A train moving east at 30 m/s seen from a train moving east at 20 m/s appears to move east at 10 m/s. The same train seen from a train moving west at 20 m/s appears to move east at 50 m/s. In two dimensions, such as a boat crossing a river, the boat's velocity relative to the ground is the vector sum of its velocity through the water and the river's current.

Where you see it

Relative velocity is everywhere in transport. A passenger walking forward inside a moving train moves faster relative to the ground than the train itself, and a passenger walking backward moves slower. Rain appears to slant toward a moving car because the car's motion adds a horizontal component to the rain's apparent velocity. An aeroplane's ground speed is its airspeed plus or minus the wind, which is why flights take longer flying into a headwind. Crossing a river, a swimmer must aim upstream so that the current carries them straight across — the swimmer's path is set by the vector sum of the two velocities.

Common mistakes

• Always adding velocities. Correction: subtract when objects move in the same direction; add only when they move toward each other.
• Forgetting that direction matters. Correction: relative velocity is a vector — a negative result means motion in the opposite direction to the one chosen as positive.
• Mixing up the frame of reference. Correction: always say relative to what — ground, train, water — or the number is meaningless.

Practice questions

1. Car A travels east at 30 m/s and car B travels east at 20 m/s. What is A's velocity relative to B?
   Show answer

   v = 30 - 20 = 10 m/s east.

2. Two trains approach each other, one at 25 m/s and the other at 15 m/s. At what speed do they approach?
   Show answer

   25 + 15 = 40 m/s — they close the gap at 40 m/s.

3. A boat moves at 4 m/s across a river whose current is 3 m/s. What is the boat's speed relative to the bank?
   Show answer

   The velocities are at right angles, so speed = √(4² + 3²) = √25 = 5 m/s.

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