Position, Velocity, and Acceleration

"Change in position" refers to the **displacement** of an object. It is a vector quantity representing the straight-line distance and direction from an object's initial position to its final position. Displacement is calculated as Final Position minus Initial Position.

In calculus, velocity is the instantaneous rate of change of position with respect to time (the first derivative of the position function), and acceleration is the instantaneous rate of change of velocity with respect to time (the first derivative of velocity or the second derivative of position).

Conversely, velocity is the integral of acceleration, and position is the integral of velocity.

Assuming **constant acceleration**, the change in position (displacement) can be found using kinematic equations. The most direct formula relating initial velocity, acceleration, and time to displacement is:

Δs = v₀t + ½at²

This formula is derived by integrating the velocity function (v(t) = v₀ + at) with respect to time.

Besides the primary formula, other common kinematic equations for constant acceleration are:

• v = v₀ + at (Final velocity)
• v² = v₀² + 2aΔs (Final velocity squared, useful when time is unknown)
• Δs = ½(v₀ + v)t (Displacement based on average velocity)

You can use these in combination to find the change in position depending on which variables are given.

If acceleration is not constant (i.e., it is a function of time, velocity, or position), you must use calculus:

1. If you have the acceleration function a(t), integrate it with respect to time to find the velocity function v(t). You'll need an initial condition (like initial velocity, v₀) to find the constant of integration.
2. Once you have the velocity function v(t), integrate it over the desired time interval [t₁, t₂] to find the change in position (displacement):
   Δs = ∫t1t2 v(t) dt

If you have velocity or acceleration as functions of other variables (like velocity as a function of position), solving can involve differential equations or other integration techniques.