Falling objects form an interesting class of motion problems. For example, we can estimate the depth of a vertical mine shaft by dropping a rock into it and listening for the rock to hit the bottom. By applying the kinematics developed so far to falling objects, we can examine some interesting situations and learn much about gravity in the process.
The most remarkable and unexpected fact about falling objects is that, if air resistance and friction are negligible, then in a given location all objects fall toward the center of Earth with the same constant acceleration, independent of their mass. This experimentally determined fact is unexpected, because we are so accustomed to the effects of air resistance and friction that we expect light objects to fall slower than heavy ones.
A hammer and a feather will fall with the same constant acceleration if air resistance is considered negligible. This is a general characteristic of gravity not unique to Earth, as astronaut David R. Scott demonstrated on the Moon in 1971, where the acceleration due to gravity is only
In the real world, air resistance can cause a lighter object to fall slower than a heavier object of the same size. A tennis ball will reach the ground after a hard baseball dropped at the same time. (It might be difficult to observe the difference if the height is not large.) Air resistance opposes the motion of an object through the air, while friction between objects—such as between clothes and a laundry chute or between a stone and a pool into which it is dropped—also opposes motion between them. For the ideal situations of these first few chapters, an object falling without air resistance or friction is defined to be in free-fall.
The force of gravity causes objects to fall toward the center of Earth. The acceleration of free-falling objects is therefore called theacceleration due to gravity. The acceleration due to gravity is constant, which means we can apply the kinematics equations to any falling object where air resistance and friction are negligible. This opens a broad class of interesting situations to us. The acceleration due to gravity is so important that its magnitude is given its own symbol,
One-Dimensional Motion Involving Gravity
The best way to see the basic features of motion involving gravity is to start with the simplest situations and then progress toward more complex ones. So we start by considering straight up and down motion with no air resistance or friction. These assumptions mean that the velocity (if there is any) is vertical. If the object is dropped, we know the initial velocity is zero. Once the object has left contact with whatever held or threw it, the object is in free-fall. Under these circumstances, the motion is one-dimensional and has constant acceleration of magnitude
A person standing on the edge of a high cliff throws a rock straight up with an initial velocity of 13.0 m/s. The rock misses the edge of the cliff as it falls back to earth. Calculate the position and velocity of the rock 1.00 s, 2.00 s, and 3.00 s after it is thrown, neglecting the effects of air resistance.
Draw a sketch.
We are asked to determine the position
Since we are asked for values of position and velocity at three times, we will refer to these as
Solution for Position
1. Identify the knowns. We know that
2. Identify the best equation to use. We will use
3. Plug in the known values and solve for
The rock is 8.10 m above its starting point at
Solution for Velocity
1. Identify the knowns. We know that
2. Identify the best equation to use. The most straightforward is
3. Plug in the knowns and solve.
The positive value for
Solution for Remaining Times
|Time, t||Position, y||Velocity, v||Acceleration, a|
Graphing the data helps us understand it more clearly.
Vertical position, vertical velocity, and vertical acceleration vs. time for a rock thrown vertically up at the edge of a cliff. Notice that velocity changes linearly with time and that acceleration is constant. Misconception Alert! Notice that the position vs. time graph shows vertical position only. It is easy to get the impression that the graph shows some horizontal motion—the shape of the graph looks like the path of a projectile. But this is not the case; the horizontal axis is time, not space. The actual path of the rock in space is straight up, and straight down.Discussion
The interpretation of these results is important. At 1.00 s the rock is above its starting point and heading upward, since
A simple experiment can be done to determine your reaction time. Have a friend hold a ruler between your thumb and index finger, separated by about 1 cm. Note the mark on the ruler that is right between your fingers. Have your friend drop the ruler unexpectedly, and try to catch it between your two fingers. Note the new reading on the ruler. Assuming acceleration is that due to gravity, calculate your reaction time. How far would you travel in a car (moving at 30 m/s) if the time it took your foot to go from the gas pedal to the brake was twice this reaction time?
What happens if the person on the cliff throws the rock straight down, instead of straight up? To explore this question, calculate the velocity of the rock when it is 5.10 m below the starting point, and has been thrown downward with an initial speed of 13.0 m/s.
Draw a sketch.<figure id="import-auto-id2150750" style="width: 810px;">
Since up is positive, the final position of the rock will be negative because it finishes below the starting point at
1. Identify the knowns.
2. Choose the kinematic equation that makes it easiest to solve the problem. The equation
3. Enter the known values
where we have retained extra significant figures because this is an intermediate result.
Taking the square root, and noting that a square root can be positive or negative, gives
The negative root is chosen to indicate that the rock is still heading down. Thus,
Note that this is exactly the same velocity the rock had at this position when it was thrown straight upward with the same initial speed. (See Example and Figure(a).) This is not a coincidental result. Because we only consider the acceleration due to gravity in this problem, the speed of a falling object depends only on its initial speed and its vertical position relative to the starting point. For example, if the velocity of the rock is calculated at a height of 8.10 m above the starting point (using the method fromExample) when the initial velocity is 13.0 m/s straight up, a result of
(a) A person throws a rock straight up, as explored in Example. The arrows are velocity vectors at 0, 1.00, 2.00, and 3.00 s. (b) A person throws a rock straight down from a cliff with the same initial speed as before, as in Example. Note that at the same distance below the point of release, the rock has the same velocity in both cases.</figcaption> </figure>
Another way to look at it is this: In Example, the rock is thrown up with an initial velocity of
The acceleration due to gravity on Earth differs slightly from place to place, depending on topography (e.g., whether you are on a hill or in a valley) and subsurface geology (whether there is dense rock like iron ore as opposed to light rock like salt beneath you.) The precise acceleration due to gravity can be calculated from data taken in an introductory physics laboratory course. An object, usually a metal ball for which air resistance is negligible, is dropped and the time it takes to fall a known distance is measured. See, for example, Figure. Very precise results can be produced with this method if sufficient care is taken in measuring the distance fallen and the elapsed time.<figure class="ui-has-child-figcaption" id="import-auto-id4097254" style="width: 810px;">
Positions and velocities of a metal ball released from rest when air resistance is negligible. Velocity is seen to increase linearly with time while displacement increases with time squared. Acceleration is a constant and is equal to gravitational acceleration.</figcaption> </figure>
Suppose the ball falls 1.0000 m in 0.45173 s. Assuming the ball is not affected by air resistance, what is the precise acceleration due to gravity at this location?
Draw a sketch.<figure id="import-auto-id4051158" style="width: 810px;">
We need to solve for acceleration
1. Identify the knowns.
2. Choose the equation that allows you to solve for
3. Substitute 0 for
4. Substitute known values yields
The negative value for
A chunk of ice breaks off a glacier and falls 30.0 meters before it hits the water. Assuming it falls freely (there is no air resistance), how long does it take to hit the water?
We know that initial position
where we take the positive value as the physically relevant answer. Thus, it takes about 2.5 seconds for the piece of ice to hit the water.
Learn about graphing polynomials. The shape of the curve changes as the constants are adjusted. View the curves for the individual terms (e.g.
- An object in free-fall experiences constant acceleration if air resistance is negligible.
- On Earth, all free-falling objects have an acceleration due to gravity
g, which averages g=9.80 m/s2.
- Whether the acceleration a should be taken as
+gor −gis determined by your choice of coordinate system. If you choose the upward direction as positive, a=−g=−9.80 m/s2is negative. In the opposite case, a=+g=9.80 m/s2is positive. Since acceleration is constant, the kinematic equations above can be applied with the appropriate +gor −gsubstituted for a.
- For objects in free-fall, up is normally taken as positive for displacement, velocity, and acceleration.
- the state of movement that results from gravitational force only
- acceleration due to gravity
- acceleration of an object as a result of gravity