Relative velocity describes how the motion of an object looks from a chosen frame of reference. A boat crossing a river, a plane flying through wind, and a person walking on a moving train all have velocities that depend on who is observing. This matters because real motion often combines an object's own motion with the motion of the medium or platform carrying it.
Vector addition lets us predict the actual path seen from the ground.
Understanding Physics: Relative Velocity
A frame of reference is more than a place where somebody stands. It includes the coordinate directions and the clock used to measure motion. A passenger sitting in a train has a useful train frame.
A person on the platform has a useful ground frame. Neither description is wrong. They answer different practical needs.
In ordinary school problems, frames move at constant velocity relative to one another. This makes the relationship between measurements simple.
If one frame speeds up, slows down, or turns, the analysis becomes harder because it is a non-inertial frame. Passengers then feel effects such as being pushed sideways during a turn, even when no sideways contact force is visible.
The safest method is to choose one reference frame as the starting point and draw every velocity from that frame. Break each velocity into horizontal and vertical parts. Give each direction a sign before doing any arithmetic.
For example, choose east as positive and west as negative. A car travelling east can have a positive velocity, while a car travelling west has a negative velocity. Subtraction is especially important when two objects move in the same direction.
A faster cyclist may approach a slower cyclist at only a small speed, even though both travel quickly relative to the road. This is why passing on a motorway can feel less dramatic than the speeds shown on each vehicle's speedometer.
River problems contain two separate decisions. The boat has a speed through the water set by its engine or rowing. The river current carries the boat downstream at the same time.
If the boat is pointed straight across, it will usually land downstream from its starting point. Its crossing time depends on the part of its motion directed across the river. Its downstream drift depends on the current acting during that time.
To arrive directly opposite, the boat must aim upstream. Some of its through-water velocity cancels the current, leaving the remaining part to carry it across.
Aircraft navigation works in the same way. Pilots select a heading that may differ from the desired track over the ground because wind changes the path.
Diagrams prevent many common errors. Draw velocity arrows head to tail, with each arrow labelled by the two objects or frames it connects. Do not add speeds unless the motions are along the same line.
Perpendicular motions require components, then a resultant size found from the square root of the sum of the squared components. The direction matters just as much as the size. A calculator can produce an angle, but the signs of the components tell which quadrant the arrow belongs in.
Relative velocity appears in radar tracking, sports, sailing, traffic safety, and video games. In each case, the important idea is to state clearly who measures the motion and relative to what.
Key Facts
- Relative velocity formula: v_A/C = v_A/B + v_B/C.
- To find how A moves relative to B, use v_A/B = v_A/G - v_B/G.
- Velocity is a vector, so both magnitude and direction must be included.
- For perpendicular components, resultant speed is v = sqrt(v_x^2 + v_y^2).
- Direction from the x-axis can be found with theta = tan^-1(v_y/v_x).
- In river and wind problems, ground velocity equals object-through-fluid velocity plus fluid velocity.
Vocabulary
- Relative velocity
- Relative velocity is the velocity of one object as measured from the reference frame of another object.
- Reference frame
- A reference frame is the viewpoint or coordinate system from which position and motion are measured.
- Velocity vector
- A velocity vector describes both how fast an object moves and the direction it moves.
- Resultant velocity
- Resultant velocity is the total velocity found by adding two or more velocity vectors.
- Component
- A component is the part of a vector along a chosen axis, such as east-west or north-south.
Common Mistakes to Avoid
- Adding speeds without directions, because relative velocity requires vector addition, not simple scalar arithmetic.
- Using the wrong reference frame, because v_boat/water is not the same as v_boat/ground and leads to a different path.
- Forgetting the current or wind velocity, because the moving fluid changes the ground velocity even when the object aims straight across.
- Using tan^-1 with the wrong component ratio, because swapping v_x and v_y gives the complementary angle instead of the actual direction.
Practice Questions
- 1 A boat can move at 4.0 m/s relative to still water and points straight north across a river flowing east at 3.0 m/s. Find the boat's speed relative to the ground and the angle of its path east of north.
- 2 An airplane has an airspeed of 120 m/s due east while a wind blows 40 m/s due north. Find the plane's ground speed and direction relative to east.
- 3 A passenger walks toward the back of a train while the train moves forward at constant speed. Explain how the passenger's velocity relative to the train can differ from the passenger's velocity relative to the ground.