Is the Velocity of a Point Constant or Variable?

[Jack3145]

Homework Statement

A point: $$[x_{0}, y_{0}, z_{0}, t_{0}]$$
It's velocity: $$V_{a} = [v_{1}, v_{2}, v_{3}, v_{4}]$$

What is wrong with this equation:

$$x^{b} = [x_{0} + v_{1} * t, y_{0} + v_{2} * t, z_{0} + v_{3} * t, t_{0} + t]$$

 

[CompuChip]

Nothing is wrong with that, as long as V is a constant vector.
For example, if V_a = (2, 3, 4, 1) then your formula for x satisfies v = dx/dt.
However, if V_a = (3t, t², 3t - 12, t^2/6) it doesn't.

 

[nlightner]

The velocity of a point is a variable, not a constant. It can change depending on the point's position and the direction in which it is moving. Therefore, the equation x^{b} = [x_{0} + v_{1} * t, y_{0} + v_{2} * t, z_{0} + v_{3} * t, t_{0} + t] is incorrect because it assumes that the velocity remains constant over time, which is not always the case. A more accurate equation would be x^{b} = [x_{0} + v_{1} * t_{b}, y_{0} + v_{2} * t_{b}, z_{0} + v_{3} * t_{b}, t_{0} + t_{b}], where t_{b} is the time at which the point's position is being measured. This takes into account that the velocity can change over time and should be calculated at the specific time t_{b}.