Motion in Space: Velocity and Acceleration

The motion of a drone in flight can be described by a vector-valued function r(t).

For each time t, r(t) gives the position vector of the drone.

To analyze its movement, observe how the position vector changes over time.

The average velocity is found by dividing the change in position by the time interval.

As the time interval becomes very small, the average velocity approaches a limit. This limit is the derivative of r with respect to t.

This derivative is the velocity vector, with each component differentiated separately. This vector points in the drone’s instantaneous direction of motion.

The velocity vector is tangent to the path and shows both direction and rate of motion.

The speed is the magnitude of this velocity vector, a scalar quantity.

Acceleration is the derivative of velocity, showing how the drone's speed or direction changes.

For example, if the position vector is t cubed i plus t squared j plus 2 k, then the velocity vector is the derivative of r(t), which equals 3t squared i plus 2t j.

Taking the derivative of velocity gives the acceleration vector: 6t i plus 2 j, showing how motion evolves in space.