04Classical Mechanics

Free fall

Drop a body from rest and, if the air does not get in the way, it falls with the same constant acceleration \(g\) whatever its size, shape or mass. Free fall is therefore a variant of the invariant: start from rest and take \(a = g\). The course writes it with the height fallen \(H\) and the speed \(V\), both measured downwards, and that is the notation used here; type any of them, or the height of a building, and the laboratory finds the rest.

Laboratory

\(H(t)\) grows as \(t^2\), \(V(t)\) as \(t\) and the acceleration is the same at every instant. Drag over a graph to travel through the fall.

The invariant

With \(V_o = 0\) and \(a = g\) the invariant keeps four quantities, \(H\), \(V\), \(t\) and \(g\), related by four equations, each of which leaves one of them out. With \(g\) known, a single datum fixes the whole fall.

Step-by-step solution

Notable events

Moments of the fall that statements often ask about.

Theory

Every body falls alike

If a body falls in a vacuum, so that the air does not affect it, a remarkable fact appears: all bodies, whatever their size, shape or composition, fall with the same acceleration in the same region near the surface of the Earth. This free-fall acceleration \(g\) is about \(9.8\ \text{m/s}^2\): the velocity of a falling body grows by 9.8 m/s every second.

It also acts on bodies moving up: whether the velocity points up or down, the acceleration due to gravity always points down.

The variant of free fall

Starting from the invariant \(V = V_o + at\), \(V^2 = V_o^2 + 2aS\), \(S = V_o t + \tfrac{at^2}{2}\) and applying the conditions of the case, release from rest (\(V_o = 0\)) and \(a = g\), with the distance \(S = H\) measured downwards:

\[V(t) = g\,t,\qquad V(t)^2 = 2g\,H,\qquad H(t) = \frac{g\,t^2}{2}.\]

The acceleration is taken as positive because the speed increases, as the course notes indicate.

Four equations, four quantities

The fall involves \(H\), \(V\), \(t\) and \(g\), and each equation leaves one of them out: \(V = gt\) has no \(H\), \(V^2 = 2gH\) has no \(t\), \(H = \tfrac12 g t^2\) has no \(V\), and \(H = \tfrac12 V t\) has no \(g\). On Earth \(g\) is known, so one datum is enough; on another planet two data are needed and the fall measures \(g\).

Galileo and the odd numbers

Galileo could not time a free fall directly, so he slowed it down with inclined planes and found that the distances covered in successive equal intervals of time grow as the odd numbers \(1 : 3 : 5 : 7\ldots\), which is the same as saying that the total distance grows as \(t^2\). It follows from \(H = \tfrac12 g t^2\): after 1, 2, 3, 4 intervals the distances are in the ratio \(1 : 4 : 9 : 16\), and their differences are the odd numbers.

Up or down: the sign of g

Resnick takes the \(y\) axis upwards; then the acceleration is \(-g\) (with \(g\) always positive) and a falling body has negative displacement and velocity: \(y = y_0 - \tfrac12 g t^2\), \(v = -gt\). The course measures \(H\) and \(V\) downwards and writes them positive. Both pictures are equivalent; what matters is to choose one and keep it during the whole problem.

Motions that happen at the same time

While a body falls, everything else keeps moving according to its own law, and the time is what links them. A cowboy that drops from a branch onto a galloping horse falls 3.00 m in \(t = \sqrt{2H/g} = 0.782\ \text{s}\); in that time a horse at 10.0 m/s advances \(d = u\,t = 7.82\ \text{m}\). This independence of simultaneous motions is the key to projectile motion.

Measuring g

The value of \(g\) changes with latitude and altitude (from about \(9.78\) at the equator to \(9.83\ \text{m/s}^2\) at the poles) and with the density of the crust below. The standard value is \(g_n = 9.80665\ \text{m/s}^2\). It can be measured timing a free fall with photogates, with a pendulum, or by throwing a body up and timing its passage through two levels.

The air, ignored

All of this assumes that air resistance is negligible, which is reasonable for a stone falling a few metres but not for a feather or for a parachutist. With air the acceleration decreases as the speed grows, until a terminal speed is reached; that case needs forces that depend on velocity and is studied with Newton’s laws.

Problem bank

Problems adapted from Serway (chapters 2, 4, 5 and 6), Resnick and the course notes. Load any of them into the laboratory: the data fill the panel and the solver does the rest, so you can then change any value and see how the answer reacts.

  1. Positions second by second

    Resnick, sample problem 2-5

    A body is released from rest. Find its position and velocity after \(1.0\ \text{s}\). Then use "query an instant" for 2.0, 3.0 and 4.0 s.

  2. A daring cowboy

    Serway 2.43 · class notes

    A daring cowboy sitting on a tree branch wants to drop vertically onto a horse galloping under the tree. The constant speed of the horse is \(10.0\ \text{m/s}\) and the distance from the branch to the saddle is \(3.00\ \text{m}\). (a) What must the horizontal distance between saddle and branch be when he makes his move? (b) For how long is he in the air?

  3. From the top of a building

    Own example

    A stone is dropped from the top of a building \(45.0\ \text{m}\) high. How long does it take to reach the ground and with what speed does it hit it?

  4. Catching a bill

    Serway 2.40

    Emily holds a dollar bill vertically with its centre between David’s index finger and thumb; David must catch it after she lets it go, without moving his hand down. If his reaction time is \(0.2\ \text{s}\), will he succeed? (The bill is about \(15.5\ \text{cm}\) long, so its upper edge is \(7.8\ \text{cm}\) above his fingers.)

  5. A cat rolls off a TV

    Serway 2.36

    A sleeping cat rolls gently off the top of a warm TV set. Ignoring air resistance, find its position and velocity after \(0.100\ \text{s}\); then query 0.200 s and 0.300 s.

  6. How high for 20 m/s?

    Own example

    From what height must a body be dropped to reach the ground at \(20.0\ \text{m/s}\)? How long does the fall last?

  7. Free fall on another planet

    Own example

    An astronaut drops a hammer from a height of \(10.0\ \text{m}\) and it takes \(3.51\ \text{s}\) to reach the ground. What is the acceleration of gravity there, and with what speed does it arrive? (Mark "g is unknown".)

Theory after R. Resnick, D. Halliday and K. Krane, Física, vol. 1, chapter 2 (sections 2-7 to 2-9); problems from R. Serway and J. Jewett, Física para ciencias e ingeniería, vol. 1, chapter 2, and from the course notes.