AcademyKinematics
Academy
Acceleration
Level 1 - Math II (Physics) topic page in Kinematics.
Principle
Acceleration measures how velocity changes with time. It can change the speed of a particle, the direction of its velocity, or both.
Instantaneous acceleration is the derivative of velocity. Since velocity is the derivative of position, acceleration is also the second derivative of position.
Notation
\(\mathbf v(t)\)
instantaneous velocity
\(\mathrm{m\,s^{-1}}\)
\(\mathbf a(t)\)
instantaneous acceleration
\(\mathrm{m\,s^{-2}}\)
\(\Delta\mathbf v\)
change in velocity
\(\mathrm{m\,s^{-1}}\)
\(\Delta t\)
elapsed time
\(\mathrm{s}\)
\(\ddot x, \ddot y, \ddot z\)
second time derivatives of position components
\(\mathrm{m\,s^{-2}}\)
Method
Step 1: Average over a time interval
Average acceleration is change in velocity divided by elapsed time:
Average acceleration
\[\mathbf a_{\text{avg}}=\frac{\Delta\mathbf v}{\Delta t}\]
Step 2: Differentiate for instantaneous acceleration
Take the time derivative of velocity:
Instantaneous acceleration
\[\mathbf a=\frac{d\mathbf v}{dt}=\frac{d^2\mathbf r}{dt^2}\]
Step 3: Work component by component
In a fixed Cartesian basis, each acceleration component is the derivative of the corresponding velocity component.
Position components
\[\mathbf r=x\mathbf i+y\mathbf j+z\mathbf k\]
Velocity components
\[\mathbf v=\dot x\mathbf i+\dot y\mathbf j+\dot z\mathbf k\]
Differentiate velocity
\[\mathbf a=\ddot x\mathbf i+\ddot y\mathbf j+\ddot z\mathbf k\]
Rules
Component acceleration
\[\mathbf a=\ddot x\mathbf i+\ddot y\mathbf j+\ddot z\mathbf k\]
Constant acceleration velocity
\[\mathbf v(t)=\mathbf v_0+\mathbf a t\]
Constant acceleration position
\[\mathbf r(t)=\mathbf r_0+\mathbf v_0t+\frac12\mathbf a t^2\]
The constant-acceleration formulas apply only when \(\mathbf a\) is constant in time.
Examples
Question
Velocity changes from
\[2\mathbf i\]
\[m s^{-1}\]
to \[8\mathbf i\]
\[m s^{-1}\]
in \[3\]
s. Find average acceleration.Answer
The velocity change is
\[6\mathbf i\]
\[m s^{-1}\]
Divide by \[3\]
s to get \[\mathbf a_{\text{avg}}=2\mathbf i\]
\[m s^{-2}\]
Checks
- Acceleration can be non-zero even when speed is constant, because direction may be changing.
- Use vector notation when acceleration has more than one component.
- Constant-acceleration equations are not valid for arbitrary acceleration functions.
- Check units after differentiating velocity: \(m s^{-1}\) divided by \(s\) gives \(m s^{-2}\).