B. Kinematics · Kinematics: Motion in One and Two Dimensions

Motion in a Vertical Line: Free Fall and Objects Thrown Up

B2.7 B2.3 B3.1 A1.5About 70 minutes

Learning goals

I can explain what free fall is and state the acceleration due to gravity near Earth's surface.
I can use a sign convention for vertical motion.
I can solve problems about dropped objects and objects thrown up or down.
I can explain why an object at the top of its flight has zero velocity but is still accelerating.

Words to know

free fall
Motion in which gravity is the only thing affecting an object (air resistance ignored).

acceleration due to gravity (g)
Near Earth's surface, g = 9.8 m/s² [down] for every object in free fall, whatever its mass.

air resistance
The push of the air against a moving object. It is ignored in free-fall problems unless stated.

maximum height
The highest point of a vertical throw, where the vertical velocity is momentarily zero.

airtime (time of flight)
The total time an object spends in the air.

What goes up...

Toss your keys straight up and catch them. In that one second they slow down, stop, turn around and speed up, and they do all of it with the same acceleration. Falling and rising objects are the most common example of uniformly accelerated motion, so everything from the last three lessons applies directly.

Free fall and g

An object is in free fall when gravity is the only thing affecting its motion. In a vacuum, a feather and a bowling ball fall side by side. Near Earth's surface every freely falling object has the same acceleration:

g = 9.8 m/s² [down]

(It varies slightly from place to place, from about 9.78 m/s² at the equator to about 9.83 m/s² at the poles. We use 9.8 m/s² in this course.)

In real life, air resistance slows light, spread-out objects like feathers and sheets of paper. For compact objects like balls, stones and keys moving at modest speeds, ignoring air resistance gives good answers. Why gravity produces this acceleration, and how air resistance leads to terminal velocity, are explained in the Forces unit.

Sign convention for vertical motion

Choose a direction and stick to it.

Up positive (usual for objects thrown upward): a = −9.8 m/s² for the whole flight, going up and coming down. A position below the launch point is negative.
Down positive (handy for objects that only fall): a = +9.8 m/s².

The acceleration is always 9.8 m/s² downward during free fall, at every instant, including at the very top.

Worked example

Example 1: A dropped stone

A stone is dropped from a bridge and hits the water 2.5 s later. Ignoring air resistance, find (a) the height of the bridge above the water and (b) the stone's velocity just before it hits.

Given: down positive; v₁ = 0 (dropped), a = +9.8 m/s², Δt = 2.5 s. Required: Δd and v₂.
(a) v₂ is not given: Δd = v₁Δt + ½aΔt² = 0 + ½(9.8)(2.5)² = 30.6 m ≈ 31 m.
(b) v₂ = v₁ + aΔt = 0 + (9.8)(2.5) = 24.5 m/s, so v₂ ≈ 25 m/s [down].
Check: Δd = ½(0 + 24.5)(2.5) = 30.6 m ✓.

(a) about 31 m; (b) about 25 m/s [down]

Worked example

Example 2: A ball thrown straight up

A ball is thrown straight up at 14.7 m/s and caught at the same height. Find (a) the time to reach the top, (b) the maximum height, (c) the total time in the air, and (d) the velocity and position at t = 2.0 s.

Up positive: v₁ = +14.7 m/s, a = −9.8 m/s².
(a) At the top, v₂ = 0. Δt = (v₂ − v₁) ÷ a = (0 − 14.7) ÷ (−9.8) = 1.5 s.
(b) Δd = (v₂² − v₁²) ÷ (2a) = (0 − 216.09) ÷ (−19.6) = 11.0 m ≈ 11 m above the launch point.
(c) Coming down to the same height takes as long as going up (the motion is symmetrical), so total airtime = 2 × 1.5 = 3.0 s. Check with Δd = 0: 0 = 14.7Δt − 4.9Δt² gives Δt = 0 or 3.0 s ✓.
(d) v₂ = 14.7 + (−9.8)(2.0) = −4.9 m/s, so 4.9 m/s [down]. Δd = (14.7)(2.0) + ½(−9.8)(2.0)² = 29.4 − 19.6 = 9.8 m above the launch point (on the way down).

(a) 1.5 s; (b) 11 m; (c) 3.0 s; (d) 4.9 m/s [down], 9.8 m up

Example 2 every 0.5 s (heights to scale, up positive). The velocity arrows shrink by 4.9 m/s every half second, reach zero at the top, then grow downward. The acceleration arrow is the same at every instant, including at the top.

The v–t graph of the throw is one straight line with slope −9.8 m/s². It crosses zero at the top of the flight. The area above the axis (going up) equals the area below it (coming down), so the ball returns to its launch height.

Worked example

Example 3: Thrown downward

A student on a balcony throws a ball straight down at 5.0 m/s from 20.0 m above the ground. Find (a) its speed just before it hits the ground and (b) the time it takes.

Down positive: v₁ = +5.0 m/s, a = +9.8 m/s², Δd = +20.0 m.
(a) Δt not given: v₂² = v₁² + 2aΔd = 25 + 2(9.8)(20.0) = 25 + 392 = 417, so v₂ = √417 ≈ 20.4 m/s, about 20 m/s [down].
(b) Δt = (v₂ − v₁) ÷ a = (20.42 − 5.0) ÷ 9.8 ≈ 1.6 s (using the unrounded v₂).

(a) about 20 m/s [down]; (b) about 1.6 s

Just for fun

In 1971, Apollo 15 astronaut David Scott stood on the Moon, where there is no air, and dropped a hammer and a falcon feather at the same moment. They hit the ground together. Galileo would have been thrilled. (The Moon's gravity is weaker than Earth's, so both fell more slowly than they would here, but they still fell together.)

Hands-on activity

Measure g with a phone camera

You need: a tennis ball or other small, heavy-ish ball, a measuring tape, a phone that can record slow-motion video, and a partner. Grown-up note: if you drop from a height, stand on a stable surface such as the top of a staircase landing with someone spotting; never lean over railings or use a ladder alone.

Part A (air resistance): drop a coin and a flat sheet of paper from the same height at the same time. Then crumple the paper into a tight ball and repeat. What changed?

Part B (measuring g):

Tape a measuring tape vertically to a wall. Your partner films in slow motion from a few metres away, with the whole fall in view.
Drop the ball from rest from about 2.0 m beside the tape.
Play the video frame by frame. Count frames from release to impact and divide by the frame rate (for example, 240 frames per second) to get Δt.
Calculate g = 2Δd ÷ Δt². Repeat three times and average.
Find your percent error: |your value − 9.8| ÷ 9.8 × 100%.

What to look for: in Part A, the flat paper floats down slowly but the crumpled paper falls almost with the coin, so shape (air resistance), not mass, made the difference. In Part B, a 2.0 m drop takes about √(2 × 2.0 ÷ 9.8) ≈ 0.64 s, about 153 frames at 240 frames per second. Values within about 5% of 9.8 m/s² are a good result; errors come from judging the exact release and impact frames.

Your turn

A pebble is dropped down a well and hits the water 1.8 s later. How deep is the well (to the water)?

How fast is the pebble moving when it hits the water?

A ball is thrown straight up at 9.8 m/s. How long does it take to reach the top?

How high does that ball go?

At the top of its flight, what are the ball's velocity and acceleration?

A ball is thrown up at 12 m/s and caught at the same height. What is its velocity when caught (ignore air resistance)?

More practice: warm-up

What is the value of g near Earth's surface?

With up as positive, what is a for a ball thrown upward?

A dropped object starts with what velocity?

How fast is a dropped object falling after 3.0 s?

In a vacuum, which hits the ground first: a feather or a hammer dropped together?

More practice: on your own

A ball is thrown up at 19.6 m/s. Find its maximum height and total airtime (caught at the launch height).

A rock is thrown down at 8.0 m/s from a 30 m cliff. How fast does it hit the bottom?

How long does the rock in the last question take to fall?

A ball thrown upward is at 6.0 m above the launch point moving up at 4.0 m/s. What was its launch speed?

Explain why a sheet of paper falls more slowly than a coin, but a crumpled sheet falls almost as fast as the coin.

Challenge

A ball is thrown up at 10.0 m/s from the edge of a cliff 25.0 m above the water. How long is it in the air before hitting the water?

A ball is dropped from rest. Show that it falls 4 times as far in 2.0 s as in 1.0 s.

From a window, a ball is thrown straight up at 5.0 m/s and another is thrown straight down at 5.0 m/s at the same moment. Compare their speeds when they hit the ground. Explain.

Check your understanding

Quiz

Near Earth's surface, ignoring air resistance, a falling object's acceleration is:

At the highest point of a straight-up throw, the ball's velocity and acceleration are:

A stone is dropped from rest. After 2.0 s it has fallen:

A ball is thrown up at 29.4 m/s. It reaches the top after:

With up as positive, the v–t graph of a ball thrown upward is:

Why does a sheet of paper fall more slowly than a coin?

For parents and teachers

Big idea: Free fall is uniformly accelerated motion with a = 9.8 m/s² [down], so it uses the same equations (B2.3, B2.7). The big conceptual point is that the acceleration does not change at the top of the flight even though the velocity is zero (B3.1). The phone-video activity is a real investigation with percent error (A1.5).

How to help: Toss a ball up and ask: "Where is it slowest? What is its acceleration there?" The answers should be "at the top" and "still 9.8 m/s² down".

Common difficulties:

Saying the acceleration is zero at the top.
Mixing sign conventions in one problem (using +9.8 with up positive).
Thinking heavier objects fall faster (that is an air-resistance effect, not gravity).
Forgetting that a position below the launch point is negative when up is positive.