Notes
Describing position
- Distance is the total path length (scalar). Displacement $\Delta x = x_f - x_i$ is the change in position (vector) and can be negative or zero.
- Pick an origin and a positive direction before solving.
Speed, velocity and acceleration
- Average speed $\displaystyle = \frac{\text{distance}}{\text{time}}$; average velocity $\displaystyle = \frac{\Delta x}{\Delta t}$.
- Acceleration $\displaystyle a = \frac{\Delta v}{\Delta t}$, in $\text{m/s}^2$. Speeding up, slowing down and turning are all acceleration.
- Convert km/h to m/s by dividing by 3.6.
Motion graphs
- Position-time graph: slope = velocity. Velocity-time graph: slope = acceleration, area = displacement.
Kinematic equations (constant acceleration)
- $v_f = v_i + at$, $\displaystyle \Delta x = v_i t + \frac{1}{2}at^2$, $v_f^2 = v_i^2 + 2a\Delta x$, $\displaystyle \Delta x = \frac{v_i + v_f}{2}t$.
- Free fall: $a = 9.8 \text{ m/s}^2$ downward for every object when air resistance is ignored.
Topics in this unit
Each topic has its own short notes and practice.
Practice
Take east as positive: $3 - 7 = -4 \text{ km}$, which is 4 km west.
$\displaystyle \frac{180}{2.5} = 72 \text{ km/h}$.
Velocity times time gives displacement.
$\displaystyle \Delta x = \frac{1}{2}(2.5)(4^2) = 20 \text{ m}$.
At the top $v = 0$: $0 = 19.6 - 9.8t$, so $t = 2 \text{ s}$.
All objects in free fall have the same acceleration, $9.8 \text{ m/s}^2$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A car waits at a red light. When the light turns green, it accelerates uniformly from rest at $3.0 \text{ m/s}^2$ for 8.0 s, then drives at constant velocity for 10 s.
(a) What is the car's velocity at the end of the 8.0 s of acceleration?
Show the answer and scoring guide
$v_f = v_i + at = 0 + (3.0)(8.0) = 24 \text{ m/s}$.
Check off the points you earned:
(b) How far does the car travel while accelerating?
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$\displaystyle \Delta x = v_i t + \frac{1}{2}at^2 = 0 + \frac{1}{2}(3.0)(8.0^2) = 96 \text{ m}$.
Check off the points you earned:
(c) What is the total distance traveled in the 18 s, and what is the average speed for the whole trip?
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Constant-velocity part: $24 \times 10 = 240 \text{ m}$. Total $= 96 + 240 = 336 \text{ m}$. Average speed $\displaystyle = \frac{336}{18} \approx 18.7 \text{ m/s}$.
Check off the points you earned: