S1.1 One dimensional particle kinematics
[A] Context
- We focus on the kinematics of a particle moving in one dimension (‘1D’, or ‘d=1’).
- Fundamental concern: variation of position (x) with time (t)
[B] Displacement
Consider a particle whose x coordinate varies smoothly but arbitrarily with t. |
- Suppose particle moves from position x1 at time t1 to x2 at t2.
- We define the time interval (1.1)Δt=t2-t1
- We define the associated displacement by
(1.2)Δx=x2-x1
[C] Velocity
From the variation of x with t we define the average velocity over a time interval as |
Key Point 1.1
The instantaneous velocity is defined as the average velocity over the next infinitesimally small time interval:![\[ v= \lim_{\Delta t \rightarrow 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt} \]](mastermathpng-1.png)
[D] Acceleration
Now consider the variation of v with t. The average acceleration over a time interval is defined as |
Key Point 1.2
The instantaneous acceleration is the average acceleration over the next infinitesimally small time interval:![\[ a= \lim_{\Delta t \rightarrow 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt} \]](mastermathpng-3.png)
[E] Integral forms of key equations
Key Point 1.3
The integral forms of the x-t and v-t relationships are:![\[ \Delta x= \int_{t_1}^{t_2} v dt \hspace{0.5cm}\mbox{\rm and}\hspace{0.5cm} \Delta v= \int_{t_1}^{t_2} a dt \]](mastermathpng-4.png)
[F] Constant acceleration equations
Key Point 1.4
For 1D motion at constant acceleration a, the position and velocity (x and v) at the end of a time interval (t) are related to those at the beginning of the interval (x0 and v0) by

Analysis
Let the times t1 and t2 be 0 and t respectively.
Denote the associated positions and velocities by x0, v0 and x, v
Recall definition
But for constant acceleration
ThusRearranging gives
v=v0+atNext recall the definition
But for constant accelerationCombining these two equations gives
or
Finally eliminating t between
gives DIY!
Visualization
Learning Resources
![]() | HRW Chapter 2 |
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