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Acceleration Vectors

Level 1 - Physics topic page in Motion in Space.

Principle

Acceleration is the time derivative of velocity, so it measures how the velocity vector changes.

The velocity vector can change in magnitude, direction, or both. Acceleration points with the velocity-change vector, not necessarily with the velocity itself.

Notation

\(\vec a\)
acceleration vector
\(\mathrm{m\,s^{-2}}\)
\(a_x,a_y,a_z\)
acceleration components
\(\mathrm{m\,s^{-2}}\)
\(\vec v\)
velocity vector
\(\mathrm{m\,s^{-1}}\)
\(\Delta\vec v\)
change in velocity
\(\mathrm{m\,s^{-1}}\)
\(\vec r\)
position vector
\(\mathrm{m}\)

Method

Derivation 1: Start from velocity change

Acceleration is built from the change in velocity over a time interval. The instantaneous value is the limit of that ratio.

Velocity change
\[\Delta\vec v=\vec v_f-\vec v_i\]
Average acceleration
\[\vec a_{\mathrm{avg}}=\frac{\Delta\vec v}{\Delta t}\]
Instantaneous acceleration
\[\vec a=\lim_{\Delta t\to0}\frac{\Delta\vec v}{\Delta t}=\frac{d\vec v}{dt}\]

Derivation 2: Work component by component

For Cartesian axes, the basis vectors are fixed, so differentiating a velocity vector means differentiating each component.

Write velocity components
\[\vec v(t)=v_x(t)\hat{\imath}+v_y(t)\hat{\jmath}+v_z(t)\hat{k}\]
Differentiate velocity
\[\vec a(t)=\frac{dv_x}{dt}\hat{\imath}+\frac{dv_y}{dt}\hat{\jmath}+\frac{dv_z}{dt}\hat{k}\]
Name acceleration components
\[\vec a=a_x\hat{\imath}+a_y\hat{\jmath}+a_z\hat{k}\]

Derivation 3: Connect to position

Because velocity is already the derivative of position, acceleration is the second derivative of position.

Velocity from position
\[\vec v=\frac{d\vec r}{dt}\]
Acceleration from velocity
\[\vec a=\frac{d\vec v}{dt}\]
Second derivative
\[\vec a=\frac{d^2\vec r}{dt^2}\]

The diagram below compares two velocity vectors placed tail-to-tail. The acceleration direction follows the vector change from initial velocity to final velocity.

0123456012345xyvivfΔv
Acceleration points in the direction of the velocity-change vector.

Rules

These are the compact results from the derivations above.

Acceleration vector
\[\vec a=\frac{d\vec v}{dt}\]
From position
\[\vec a=\frac{d^2\vec r}{dt^2}\]
Component form
\[\vec a=a_x\hat{\imath}+a_y\hat{\jmath}+a_z\hat{k}\]

Examples

Question
For
\[\vec v(t)=3t\hat{\imath}+(4-t^2)\hat{\jmath}\]
find
\[\vec a\]
Answer
Differentiate each component:
\[\vec a=3\hat{\imath}-2t\hat{\jmath}\,\mathrm{m\,s^{-2}}\]

Checks

  • Acceleration is not always parallel to velocity.
  • Direction change requires acceleration.
  • Constant speed can still have nonzero acceleration.
  • Work component by component.

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Position Vectors

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Projectile Models