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Motion in One Dimension Study Guide (page 3)

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Updated on Sep 26, 2011

Freely Falling Objects

Any object found in the neighborhood of the surface of the earth suffers a gravitational pull with a downward acceleration g = 9.80 m/s2. The exact value slightly varies depending on various factors such as the latitude and the mineral composition of the earth's crust, which finally affects the mass distribution of the earth. But for the purpose of most calculations, we may assume all objects fall down with a constant acceleration g = 9.80 m/s2. The direction is always downward, actually toward the center (or mass—for pedantic eyes) of the earth.

The main experimental result for the freely falling motion was obtained by Galileo hundreds of years ago and states that when we can neglect air resistance, all bodies fall at the same rate, no matter how heavy or light they are. He reached this conclusion by dropping at the same time different objects from the top of the Tower of Pisa and observing that they always hit the ground almost at the same time. If the experiment could be done so that there is no air drag, such as in a vacuum tube, the times of the falls would certainly be identical, and they would hit the ground exactly at the same time. And that is exactly what Robert Boyle (1627–1691) did when he perfected his vacuum pumps in the mid-1600s and had a golden opportunity to test Galileo's work for the first time.

This may sound counterintuitive at some point: If we drop a quarter and a sheet of paper from the same height at the same time, the quarter quickly reaches the floor, while the paper still slowly flutters through air for a period of time. But if we crush the paper into a wrinkled ball and therefore cut its air drag without changing its weight, it will hit the ground almost at the same time as the quarter.

In general, for a body moving along the vertical plane near the surface of Earth, we can apply the rules of constant acceleration motion. Considering the direction of motion is positive from the surface of Earth upward, the acceleration will be a = g = –9.80 m/s2 for an ascending body, and a = +9.80 m/s2 for a falling body. NOTE: The direction of the acceleration due to gravity is always directed toward the center of Earth.

Example

A bullet blasts from the barrel of a gun upward in the vertical direction with an initial speed of 700 m/s. Find the maximum altitude reached by this bullet and the time needed to reach it.

Solution

At the highest point of its path, the bullet reverses direction from moving upward to falling downward. Therefore, at this point, its instant velocity (and speed) should be zero. Using the formula linking square speeds to acceleration and distance, we find:

– 2 · 9.8m/s2 · h

Solving for h, we find:

h = 25,000 m = 25 km

To find the time needed to reach that altitude, we simply use:

v = v0 + a · t

or, plugging in the numbers:

0 = 700 m/s – 9.8 m/s2 · t

Solving for the time t, we find:

t = 71.43 s

Practice problems of this concept can be found at: Motion in One Ditnension Practice Questions

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