AcademyOne-Dimensional Motion
Academy
Constant-Acceleration Models
Level 1 - Physics topic page in One-Dimensional Motion.
Principle
Constant acceleration makes velocity linear in time and position quadratic in time. The model applies only across intervals where acceleration can reasonably be treated as constant.
Notation
\(x\)
position
\(\mathrm{m}\)
\(x_0\)
initial position
\(\mathrm{m}\)
\(v_x\)
velocity
\(\mathrm{m\,s^{-1}}\)
\(v_{0x}\)
initial velocity
\(\mathrm{m\,s^{-1}}\)
\(a_x\)
constant acceleration
\(\mathrm{m\,s^{-2}}\)
\(t\)
elapsed time
\(\mathrm{s}\)
Method
Start with constant rate
\[a_x=\frac{dv_x}{dt}\]
The acceleration has the same value throughout the modeled interval.
Integrate velocity change
\[v_x=v_{0x}+a_xt\]
Use velocity definition
\[v_x=\frac{dx}{dt}\]
Integrate position change
\[x=x_0+v_{0x}t+\frac{1}{2}a_xt^2\]
Eliminate time
\[v_x^2=v_{0x}^2+2a_x(x-x_0)\]
Combine the velocity-time and position-time results when time is not part of the problem.
The graph below shows the two signatures of the model: a straight velocity-time graph and a curved position-time graph.
Choose an equation by matching the unknown and the quantities actually given; the no-time equation is useful when the interval time is absent.
Rules
These are the compact results from the method above.
Velocity time
\[v_x = v_{0x} + a_xt\]
Position time
\[x = x_0 + v_{0x}t + \frac{1}{2}a_xt^2\]
Velocity position
\[v_x^2 = v_{0x}^2 + 2a_x(x - x_0)\]
Examples
Question
A particle has initial speed (v_{0x}) and constant acceleration
\[-a_x\]
How far does it travel before stopping?Answer
Use the no-time equation because the target is distance and the final velocity is known.
\[0=v_{0x}^2+2(-a_x)Delta x\]
\[Delta x= rac{v_{0x}^2}{2a_x}\]
Checks
- Use only for constant acceleration.
- Time is measured from the initial state.
- Signs follow the chosen axis.
- The no-time equation removes t.