1.7 Falling objects (Page 6/10)

Page 6 / 10

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 $y_{0} = 0$ , final position $y = −30 . 0 m$ , and $a = - g = - 9 . {80 m/s}^{2}$ . We can then use the equation $y = y_{0} + v_{0} t + \frac{1}{2} {at}^{2}$ to solve for $t$ . Inserting $a = - g$ , we obtain

\begin{array}{l} y & = & 0 + 0 - \frac{1}{2} {gt}^{2} \\ t^{2} & = & \frac{2 y}{- g} \\ t & = & \pm \sqrt{\frac{2 y}{- g}} = \pm \sqrt{\frac{2 (- 30.0 m)}{- 9.80 m {/s}^{2}}} = \pm \sqrt{6.12 s^{2}} = 2.47 s \approx 2.5 s \end{array}

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.

Phet explorations: equation grapher

Learn about graphing polynomials. The shape of the curve changes as the constants are adjusted. View the curves for the individual terms (e.g. $y = bx$ ) to see how they add to generate the polynomial curve.

Equation Grapher

Section summary

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
$g = 9 . {80 m/s}^{2} .$
Whether the acceleration a should be taken as $+ g$ or $- g$ is determined by your choice of coordinate system. If you choose the upward direction as positive, $a = - g = - 9 . 80 m {/s}^{2}$ is negative. In the opposite case, $a = +g = 9 . {80 m/s}^{2}$ is positive. Since acceleration is constant, the kinematic equations above can be applied with the appropriate $+ g$ or $- g$ substituted for $a$ .
For objects in free-fall, up is normally taken as positive for displacement, velocity, and acceleration.

Conceptual questions

What is the acceleration of a rock thrown straight upward on the way up? At the top of its flight? On the way down?

An object that is thrown straight up falls back to Earth. This is one-dimensional motion. (a) When is its velocity zero? (b) Does its velocity change direction? (c) Does the acceleration due to gravity have the same sign on the way up as on the way down?

Suppose you throw a rock nearly straight up at a coconut in a palm tree, and the rock misses on the way up but hits the coconut on the way down. Neglecting air resistance, how does the speed of the rock when it hits the coconut on the way down compare with what it would have been if it had hit the coconut on the way up? Is it more likely to dislodge the coconut on the way up or down? Explain.

If an object is thrown straight up and air resistance is negligible, then its speed when it returns to the starting point is the same as when it was released. If air resistance were not negligible, how would its speed upon return compare with its initial speed? How would the maximum height to which it rises be affected?

The severity of a fall depends on your speed when you strike the ground. All factors but the acceleration due to gravity being the same, how many times higher could a safe fall on the Moon be than on Earth (gravitational acceleration on the Moon is about 1/6 that of the Earth)?

How many times higher could an astronaut jump on the Moon than on Earth if his takeoff speed is the same in both locations (gravitational acceleration on the Moon is about 1/6 of $g$ on Earth)?

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Source: OpenStax, Sample chapters: openstax college physics for ap® courses. OpenStax CNX. Oct 23, 2015 Download for free at http://legacy.cnx.org/content/col11896/1.9

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