The Constant-Acceleration Equations
Derive the five constant-acceleration equations from a straight-line velocity-time graph, then choose the one that uses the three quantities you know and gives the one you want.
What a learner can do afterwards
- Derive v = u + at and s = (u + v)t / 2 from the velocity-time graph
- List the five quantities in a problem and pick the equation that leaves out the unwanted one
- Solve for time when the equation is quadratic in t and say what both roots mean
1 · Read
Five letters run the whole show: s for displacement, u for starting velocity, v for final velocity, a for acceleration, t for time. On a velocity time graph the slope gives v equals u plus a t, and the area of the trapezium gives s equals (u plus v) times t over 2. A bike starting at 3 metres per second with acceleration 1.5 reaches 9 metres per second after 4 seconds, since 3 plus 1.5 times 4 is 9.
List what you know, name what you want, and pick the equation that never mentions the leftover letter. To skip t, use v squared equals u squared plus 2 a s: from rest with acceleration 4 over 8 metres, v squared is 64, giving 8 metres per second. To skip v, use s equals u t plus half a t squared: with u 4, a 3 and t 2, s is 8 plus 6, which is 14 metres. To skip u, use s equals v t minus half a t squared.
A quadratic in t gives two roots, and you keep the one that fits the story. Starting from rest with acceleration 2, reaching 49 metres gives t squared equals 49, so t is 7 or negative 7, and the journey takes 7 seconds. Negative 7 is when the same curve would pass that point before the clock started. A root of t equals 0 marks the moment the clock starts.
Write the five letters on paper every time, tick the knowns, circle the want, and cross out the leftover before you choose. Then check the sign of each number against your direction.
List the five letters, pick the equation that skips the leftover, and read both roots against the story.
The rewritten readings were written for this dataset. Practice items adapted from WeBWorK Open Problem Library problems by Brandon Lostracco (Brock University) (CC BY-NC-SA 3.0).
2 · Watch
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