Physics / Unit 1: Motion in One Dimension
1.5 Kinematic Equations
Notes
When acceleration is constant, four equations link displacement $\Delta x$, initial velocity $v_i$, final velocity $v_f$, acceleration $a$ and time $t$.
- $v_f = v_i + at$
- $\displaystyle \Delta x = v_i t + \frac{1}{2}at^2$
- $v_f^2 = v_i^2 + 2a\Delta x$
- $\displaystyle \Delta x = \frac{v_i + v_f}{2}t$
- Strategy: list the known values, pick the unknown, and choose the equation that leaves out the variable you do not know or need.
- Example: a car starts from rest with $a = 3 \text{ m/s}^2$ for 4 s. $\displaystyle \Delta x = 0 + \frac{1}{2}(3)(4^2) = 24 \text{ m}$.
Practice
A bike starts at $4 \text{ m/s}$ and accelerates at $1.5 \text{ m/s}^2$ for 6 s. What is its final velocity?
$v_f = 4 + (1.5)(6) = 13 \text{ m/s}$.
A plane starts from rest and accelerates at $5 \text{ m/s}^2$ until it reaches $60 \text{ m/s}$. How far does it travel?
$\displaystyle v_f^2 = 2a\Delta x \Rightarrow \Delta x = \frac{60^2}{2(5)} = 360 \text{ m}$.
A car moving at $20 \text{ m/s}$ brakes at $-4 \text{ m/s}^2$. How far does it go before stopping?
$\displaystyle 0 = 20^2 + 2(-4)\Delta x \Rightarrow \Delta x = \frac{400}{8} = 50 \text{ m}$.
Ready for more? Take the Unit 1 test.