Motion vectors
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Motion Vectors. Displacement Vector. The position vector is often designated by . A change in position is a displacement. The displacement vector is a change in the position vector. y. x. Position and displacement are vectors. Time has a value but no direction – it’s a scalar.

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Displacement vector
Displacement Vector

  • The position vector is often designated by .

  • A change in position is a displacement.

  • The displacement vector is a change in the position vector.

y

x


Time is a scalar

Position and displacement are vectors.

Time has a value but no direction – it’s a scalar.

Vector quantities can be multiplied or divided by time using scalar multiplication.

Time is a Scalar

y

x


Velocity vector

Displacement divided by time gives the average velocity.

The displacement vector divided by time gives the average velocity vector.

A person walks from 2.0 km north of the gym to 3.0 km west of the gym in 1.5 h. What is the magnitude of average velocity?

The magnitude of displacement is

The average velocity is the magnitude of displacement divided by the time.

Velocity Vector


Tangent
Tangent

  • The average velocity becomes the instantaneous velocity for short time intervals.

  • The same is true for vectors.

  • The instantaneous velocity vector direction is tangent to the curve.

y

x


Derivative of a vector
Derivative of a Vector

  • The derivative of a vector can be computed as a derivative of the separate components.


Velocity graph

Position graph with velocity vectors

Velocity graph

Velocity Graph

y

vy

x

vx


Acceleration vector
Acceleration Vector

  • Average acceleration is the ratio of the change in velocity to the time.

  • Instantaneous acceleration is the derivative of the velocity with respect to time.

  • Both these definitions extend to vectors.


Vector equations
Vector Equations

  • Like velocity, acceleration equations can be written by components.

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