Is the acceleration due to gravity negative?
Updated
The short answer
It is negative when you take up as positive: anything in free fall then has a = −9.80 m/s², rising or falling, while the number g itself stays positive. The common slip is writing +9.80 with up as positive, or putting −9.80 into a formula that already subtracts g.
What it looks like
A ball is thrown straight up at 19.6 m/s. How long does it take to reach the top? Take up as positive and use v = v₀ + at, with v = 0 at the top.
| Wrong | Right |
|---|---|
| 0 = 19.6 + 9.80t | 0 = 19.6 + (−9.80)t |
| t = −2.00 s | t = 2.00 s |
The negative time is the clue, because the ball can't reach the top two seconds before it leaves your hand. With up as positive, gravity pulls against the ball the whole way up, so its acceleration is −9.80 m/s².
The same slip shows up in force problems. A rope lifts a 2.00 kg box with 30.0 N of tension. With up as positive, the weight is −mg = −19.6 N, so Newton's second law gives 30.0 − 19.6 = 2.00a and a = 5.20 m/s². Writing 30.0 + 19.6 = 2.00a has the weight pulling up alongside the rope.
In alai, an iPad app that reads your handwriting as you work, Ally, the tutor, would point at the 9.80 or the 19.6 in those wrong lines and ask which way is positive.
Why it happens
The letter g stands for a size, 9.80 m/s², and it's positive. The acceleration a is a signed number, and its sign comes from the direction you picked. Out of habit, it's easy to write a = 9.80 as if the two were the same thing.
Textbooks also write the free-fall equations in two ways. OpenStax College Physics 2e builds the minus sign into the formulas, as v = v₀ − gt and y = y₀ + v₀t − ½gt², and you put in g = 9.80 m/s². Other books keep v = v₀ + at and leave the sign to you. If you put −9.80 into a formula that already subtracts g, the two minus signs make a plus, and gravity ends up pointing up.
It can also feel wrong for the acceleration to be negative while the ball speeds up on the way down. With up as positive, the velocity on the way down is negative too, and a negative acceleration makes it more negative, which means faster.
How to catch it yourself
Before you put in numbers, draw a small arrow next to the problem for the direction you call positive. Then ask:
Which way is positive? Check the sign on a.
If gravity pulls against your arrow, the a you write needs a minus sign, unless your formula already has one.
Try these
Use g = 9.80 m/s².
- A stone is dropped from rest off a bridge. Taking down as positive, what is its velocity after 1.50 s?
- A ball is thrown straight up. Taking up as positive, what is its acceleration at the very top of its path?
- A ball is thrown straight up at 14.7 m/s. Taking up as positive, how high does it rise above your hand?
Answers
- v = 0 + (9.80)(1.50) = 14.7 m/s. It's positive because down is positive here.
- a = −9.80 m/s². The ball stops for an instant at the top, but gravity is still pulling it down.
- Use v² = v₀² + 2aΔy with v = 0: 0 = (14.7)² + 2(−9.80)Δy, so Δy = 11.0 m.
How alai helps
alai is a notes app for iPad and the web. In Homework help, its tutor, Ally, follows each line you write, and when the sign on a doesn't match the direction you chose, Ally points at it and asks which way is positive, so the change is yours to make. If you want to know why gravity gets a minus sign, you can hold to talk and ask Ally.