The Uniform Acceleration Equations
Using v = u + at and v squared = u squared + 2as when acceleration is steady, including picking which one fits the quantities given. The graph work becomes algebra that answers questions no graph was drawn for.
What a learner can do afterwards
- Lists the known and wanted quantities of a problem and picks the equation that links them
- Rearranges each equation to make any one of its quantities the subject
- Uses a negative acceleration for a braking or rising object and reads the sign of the answer correctly
1 · Read
When acceleration stays steady, use v equals u plus a t. Here u is the starting velocity, v is the final velocity, a is the acceleration and t is the time. Know three and find the fourth: to find time, write t equals v minus u over a.
When time is missing but distance s is known, reach for v squared equals u squared plus 2 a s. It links starting speed, final speed, acceleration and distance with no t in sight. Working out a falling stone's speed from its drop is this kind of question.
Suppose a car goes from 0 to 20 metres per second in 10 seconds, speeding up steadily. Average acceleration is the speed change divided by the time: 20 minus 0 over 10 is 2 metres per second squared. That is v equals u plus a t rearranged to solve for a.
List what you know and what you want before picking. If time is missing, use the second equation; if distance is missing, use the first. Keep signs honest: braking takes negative acceleration if forward is positive, and a ball thrown up has negative acceleration while still rising.
List knowns, pick the equation that fits, and keep the signs honest.
More to read
Motion with Constant Acceleration
When something speeds up or slows down at a steady rate, we call that uniform acceleration. Physicists use two handy equations to describe it.
The first is v = u + at. Here u is the starting speed, v is the speed at the end, a is the acceleration, and t is the time it took. If you know three of these four quantities, you can find the fourth by rearranging the equation. For example, to find time, you'd write t = (v - u) / a.
The second equation is v squared = u squared + 2as. This one is great when you are not given the time at all, but you do know the distance travelled, s. It links starting speed, final speed, acceleration, and distance without needing t.
The trick to using either equation well is to first write down everything you know and the one thing you want. If time is missing from your list, reach for the second equation instead of the first. If distance is missing, use the first equation.
Watch the sign of acceleration too. A car braking has a negative acceleration if you called the forward direction positive, so its speed goes down as time passes. A ball thrown straight up also has negative acceleration, since gravity pulls it back down even while it is still rising. Keep your directions consistent and the signs will tell you the right story.
Checking Your Understanding of Acceleration
Suppose a car's speed goes from 0 m/s up to 20 m/s in 10 seconds, with the speed increasing steadily the whole time.
Average acceleration is just how much the speed changes divided by how long that change took: a = (v - u) / t. Plugging in the numbers gives a = (20 - 0) / 10 = 2 m/s squared.
This is exactly the same relationship hiding inside v = u + at, just rearranged to solve for a instead of v. Whenever a problem gives you a starting speed, an ending speed, and a time, you can find the acceleration this way, or work backwards to find any of the other three quantities if the acceleration is the one you already know.
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2 · Watch
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Where it sits
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Optional connections
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Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.