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:[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:[E] Integral forms of key equations
Key Point 1.3
The integral forms of the x-t and v-t relationships are:[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
![\[ a_{\mbox{\rm av}} = \frac{\Delta v}{\Delta t} =\xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{ \frac{ v_2 -v_1}{t_2 -t_1}= \frac{ v -v_0}{t}} \]](mastermathpng-10.png)
But for constant acceleration
Thus![\[ a_{\mbox{\rm av}} = a \hspace*{1cm}\mbox{\rm (constant)} \]](mastermathpng-11.png)
![\[ a = \frac{ v -v_0}{t} \]](mastermathpng-12.png)
Rearranging gives
v=v0+atNext recall the definition
But for constant acceleration![\[ v_{\mbox{\rm av}} = \frac{\Delta x}{\Delta t}= \xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{\frac{ x_2 -x_1}{t_2 -t_1} =\frac{ x -x_0}{t}} \]](mastermathpng-13.png)
![\[ v_{\mbox{\rm av}} = \frac{v_0 +v}{2} = \xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{\frac{v_0+\left[ v_0 +at \right]}{2} = v_0+\frac{1}{2}at} \]](mastermathpng-14.png)
Combining these two equations gives
![\[ \frac{x -x_0}{t} = v_0+\frac{1}{2}at \]](mastermathpng-15.png)
or

Finally eliminating t between
![\[ v=v_0 + at \hspace*{1cm}\mbox{\rm and}\hspace*{1cm} x-x_0 = v_0 t +\frac{1}{2}at^2 \]](mastermathpng-17.png)
gives DIY!

Visualization
Learning Resources
| HRW Chapter 2 | |





