The way velocity appears differs depending on the observer's point of view. Let us understand relative velocity and combining velocities.
Basic Knowledge
Relative velocity is the velocity of one object as seen from another object.
If the velocity of object A is vA and that of object B is vB, the relative velocity of B as seen from A is:v_AB = vB - vA
(Subtract the observer A's velocity)
When moving in the same direction: v_AB = vB - vA(the difference becomes smaller)
When moving in opposite directions: v_AB = vB - vA(the magnitudes are added)
Combining velocities means finding the actual velocity when multiple velocities are combined. On a straight line, simple addition or subtraction is used; in two dimensions, the vector sum(parallelogram law) is used. For example, when combining velocities for a boat crossing a flowing river, combine the boat's velocity vector and the river's flow vector to find the actual direction of motion and speed.
Relative velocity (relative velocity)（The velocity of another object as seen by an observer）
v_AB = vB - vA（Formula for the relative velocity of B as seen from A）
Combining velocities（The resultant velocity obtained by adding multiple vectors）
Parallelogram law（The diagonal of the parallelogram formed by two vectors as sides is the resultant vector）
Galilean transformation（The rule for adding velocities. A coordinate transformation in classical mechanics）
Resultant velocity（The velocity obtained by combining multiple velocities）
Further Study (Background and Significance)
The concept of relative velocity appears frequently in everyday life. Cars traveling in the same direction on a highway move slowly relative to each other, while cars traveling in opposite directions have a large relative velocity and are dangerous.
When a ball is thrown inside a train, velocity measured from the ground = train velocity + ball velocity(velocity inside the train). This is combining velocities.
At speeds close to the speed of light c, the classical “rule for adding velocities” does not hold, and Einstein's special theory of relativity and its velocity-addition rule are required. However, for ordinary speeds(v ≪ c), the classical addition rule is sufficiently accurate.
Combining velocities in two dimensions requires the Pythagorean theorem and trigonometric functions. The actual speed of a boat crossing a river is √(v_boat² + v_river²) (when crossing perpendicular to the riverbank).
- Relative velocity of B as seen from A = vB - vA
- Same direction: relative velocity = difference in velocities
- Opposite directions: magnitude of relative velocity = sum of velocities
- Combining velocities = vector sum(parallelogram law)
- One dimension: add and subtract carefully, paying attention to signs
- Two dimensions: use the Pythagorean theorem or trigonometric functions
- Velocity relative to the ground = resultant velocity
Points to Note (Common Confusions)
① The relative velocity of B “as seen from A” is vB - vA. Do not reverse the order of subtraction. ② Relative velocity and resultant velocity are opposite operations. Relative velocity involves “subtracting,” while combining velocities involves “adding.” ③ In one dimension, when objects move in opposite directions, set the signs correctly before subtracting. ④ The velocity to find differs depending on whether the problem asks for the view “from the ground” or “from the train.”
Practice
- Train A travels east at 60 km/h, and train B also travels east at 40 km/h. Find the relative velocity of B as seen from A.
- Train A travels east at 60 km/h and train B travels west at 40 km/h. When they pass each other, find the relative velocity of B as seen from A.
- A boat with a speed of 4 m/s in still water attempts to cross a river flowing at 3 m/s perpendicular to the riverbank. Find the magnitude of the boat's resultant velocity as seen from the ground.