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.
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.