← Free fall

Signs in free fall: choose a direction and keep it

On a straight trail, north is the positive direction. Which velocity describes a runner moving south at 5 m/s?

Every free-fall calculation starts with one decision: which direction counts as positive. Make it once, and every sign in the problem follows from it.

Choose the positive direction first, and state it. Either choice is legal — the motion does not change, only the signs that describe it.
positive direction: up+the ball, about to be signedBefore any signs: choose the positive direction and state it. Here, up is positive.
positive direction: uprisingv positivea = −9.8 m/s²Rising: the velocity points up, so v is positive. The acceleration points down: a = −9.8m/s².
positive direction: upfallingv negativea = −9.8 m/s²Falling: the velocity points down, so v is negative. The acceleration is still a = −9.8m/s².

With up chosen as positive, the acceleration is a = −9.8 m/s² for the entire flight, whether the object is rising or falling.

positive direction: upends above the startstartendΔx positiveends below the startstartendΔx negativeDisplacement follows the same choice: ending above the start, Δx is positive; ending below,Δx is negative.

A ball is thrown straight up from a balcony and lands on the street below. Taking up as positive, give the signs of the launch velocity, the acceleration during the flight, and the displacement from launch to landing.

Launch velocity: the ball leaves moving upward, so v₀ is positive.

Acceleration: 9.8 m/s² downward for the entire flight, so a is negative the whole time.

Displacement: the ball lands below its launch point, so Δx is negative.

v₀ positive, a negative, Δx negative

positive direction: downfalling, with down chosen positivev positivea = +9.8 m/s²Choose down as positive instead and every sign flips: a = +9.8 m/s², and downward v and Δxare positive.

A stone is dropped from a pier into the water. Taking down as positive, give the signs of the stone's velocity and acceleration during the fall.

Velocity: the stone moves downward, the positive direction, so v is positive.

Acceleration: 9.8 m/s² downward, the positive direction, so a is positive.

v positive, a positive

Whichever you choose, keep it for the whole problem: one choice, every quantity signed to match.

A tennis ball is on its way up after being hit straight upward. Taking up as positive, what are the signs of its velocity and its acceleration?

A rock has been dropped from a rooftop and is falling. Taking down as positive, what are the signs of its velocity and its acceleration?

A juggling ball thrown straight up is now on its way back down. Taking up as positive, what are the signs of its velocity and its acceleration?

Your turn

A ball is thrown straight up from the edge of a cliff and lands at the base of the cliff, below the launch point. Taking up as positive, what are the signs of the launch velocity v₀, the acceleration a, and the whole-flight displacement Δx?

A beanbag is tossed straight up. This problem takes down as the positive direction. What are the signs of the launch velocity and of the acceleration while the beanbag rises?

A ball is thrown straight up and later caught at the height it was thrown from. Taking up as positive, which quantity keeps the same sign for the entire flight?