Free Fall on a Small Airless Planet A rock released from...

Free Fall on a Small Airless Planet A rock released from rest to fall on a small airless planet falls $y = g t ^ { 2 } \mathrm { m }$ in $t \mathrm { sec } , g$ a constant. Suppose that the rock falls to the bottom of a crevasse 20$\mathrm { m }$ below and reaches the bottom in 4$\mathrm { sec. }$ (a) Find the value of $g .$ (b) Find the average speed for the fall. (c) With what speed did the rock hit the bottom?

Free Fall on a Small Airless Planet A rock released from rest to fall on a small airless planet falls $y = g t ^ { 2 } \mathrm { m }$ in $t \mathrm { sec } , g$ a constant. Suppose that the rock falls to the bottom of a crevasse 20$\mathrm { m }$ below and reaches the bottom in 4$\mathrm { sec. }$

(a) Find the value of $g .$
(b) Find the average speed for the fall.
(c) With what speed did the rock hit the bottom?

Key Concepts

Uniform Acceleration

This concept describes the motion of objects when they experience a constant acceleration, such as gravity. With constant acceleration, the formulas for displacement, velocity, and time become predictable and follow specific kinematic equations. These equations form the basis for solving problems like determining the gravitational acceleration from the displacement and time of a falling object.

Average Speed

Average speed is the total distance traveled divided by the total time taken, irrespective of the variations in speed during the motion. In problems involving uniform acceleration, even though the instantaneous speed is changing, the average speed provides a useful measure for overall motion analysis.

Instantaneous Velocity

Instantaneous velocity refers to the speed of an object at a specific moment in time. For uniformly accelerated motion, it can be determined by differentiating the displacement with respect to time or by using kinematic equations. This value is crucial for understanding the exact state of motion of the object at any given point, such as the final speed upon impact.

Transcript

00:01 Question 64 asks, a rock release from rest to fall in a small airless planet falls y equals gt squared meters in t seconds and g is a constant.

00:11 And it reaches 20 meters below in t equals four seconds.

00:16 So part a has to find the g value.

00:18 And this can really be simply done by substituting what we know.

00:22 So what we're given is y equals gt squared.

00:26 We know that at t equals four seconds, and i'm going to put this in red, at 4 seconds, we are at 20 meters.

00:38 Therefore, this is 4 times 4 is 16.

00:43 16.

00:44 I'm going to write up here.

00:46 20 divided by 16 is equal to g.

00:49 And when we simplify, we are going to get 5 over 4 as our g constant right here.

01:00 Part b asks to find the average speed for the fall.

01:04 And we did this in an earlier problem, but we're going to do it again.

01:08 And again, to find the average b, we are going to do change of y over change of t.

01:14 And we know that we want to find the y at four seconds.

01:22 Therefore, we are going to take the formula that we do know, which is y is equal to 5 over 4 t squared.

01:30 And this time we do know g.

01:32 So we are going to substitute g as 5 over 4, which is just equal to 1 .25.

01:39 And going back up here, we have 1 minus 1 .25.

01:43 And the other time that we do know, and the height that we do know at the time is 0 seconds.

01:53 So on the bottom is a change of time, which is just 4 minus 0.

01:58 And when you put this into the calculator, you're going to get 20 in numerator, divide by 4 in the denominator.

02:06 Therefore, the average speed for the fall is 5 meters per second.

02:12 Oh, sorry.

02:14 I wrote feet for some reason...

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