Question 622770: In an effort to test the laws of physics, you throw a medium-sized marble vertically upwards into the air with an initial velocity of 17.4 m/s. Your hand is 1.3 m above the ground when you release the marble. Use upwards as the positive direction.
From the time the marble leaves your hand to the time it hits the ground, what is the marble's total displacement?
The choices are:
0 m
-1.3 m
16.7 m
32.1 m
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! TRICK QUESTION.
The marble goes from a position at 1.3 meters above the ground to a maximum height and then back to the ground.
If we measure position from the ground up, the initial height was +1.3 meters and the final height was 0 meters
The overall change of position is the total displacement, and it is:
final height - initial height = 0 m - 1.3 m =  m.
It is negative because upwards is the positive direction.
MORE FUN:
As further steps to the problem, you will probably have to calculate a lot of other stuff, like time to the top, time in the air, velocity as it hits the ground, total trajectory length, and who knows what else.
After release, the upwards (positive) velocity will decrease with a constant negative/downwards acceleration of  (((m/s^2}}}
So t seconds after the marble is released, its upwards velocity (in meter per second) will be given by the function
 (a linear function)
At the top of its trajectory, the marble's velocity will have decreased to zero.
We can calculate how many seconds after release that happens:
v=0 --> 9.8t=17.4 --> t=17.4/9.8= about 1.78 seconds.
The distance covered during that time is the average of initial (17.4 m/s) and final (0 m/s) velocities, multiplied by the time elapsed.
We can use the average velocity because the velocity was changing as a linear function (with a constant slope of 9.8  ).
The average velocity is  m/s (8.7 m/s), so the distance from release to top of trajectory is
 = (8.7m/s)(about 1.78 s) = about 15.45 meters.
So the marble goes up for about 1.78 seconds, reaching a height of about 15.45 meters above the hand that released it at 1.3 meters above the ground.
The marble's maximum height measured from the ground up is about
(15.45+1.3) meters = 16.75 meters.
After that, the marble travels down to the ground accelerating at 9.8 ).
The final speed in m/s will be 9.8 times the falling time, ( , in seconds.
The average speed during the fall will be half of that (because as it started the fall the marble's velocity was zero, and it was changing linearly as it fell).
So the average falling velocity ,  times the falling time, , gives you the distance covered during the fall:
 = about 16.75 m
You can calculate the falling time from there.
After that, you can calculate the final velocity as it hits the ground as  .
And of course, the total trajectory was 15.45 m +16.75 m = 32.2 m.
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