# Kinematics - projectile motion of a falling body

1. Nov 11, 2009

### Jo Turner

1. The problem statement, all variables and given/known data

Hi all - apologies now if there is a thread on this very problem, but I have searched the web and still can't seem to find something to help me.

Okay the problem: I am trying to solve a problem that has been bugging me and I think I am missing something

I have ball bearing, a rolled up paper tube, a styrofoam cup, and various measurement tools. I lay the tube on the top of the inverted styrofoam cup and the edge of a desk, and secure it. From the side then of course it looks like a right angle triangle. If I put my ball bearing in the paper tube, it rolls down the tube and falls of the edge of the table in a parabolic motion. It all seems to be a classic projectile motion calculation. Here are the measurements of the distances:

height of styrofoam cup(exactly the height where the ball bearing starts) = 0.09m
distance from edge of table (horizontal distance) = 0.335m
length of the paper tube to edge of table = 0.35m

I want to measure the horizontal component velocity of the ball bearing after it leaves the table in order to calculate the horizontal distance it travels when it hits the floor (and then we physically measure it to check).

We also need the height of the table to the floor = 0.915m

So I am using all the usual suspect equations:

x = ut + 1/2at^2
v^2 = u^2 +2as
v=s/t

the first thing I do is calculate the time it takes for the ball to roll down the paper tube to the edge of the table. given that acceleration in the vertical direction is gravity and there is no acceleration in the horizontal direction (the ball is released not pushed). in the vertical direction then:
x = ut + 1/2at^2
0.09m = 0 x t + 1/2 x 9.8 x t^2
solve for t = 0.136s

therefore in the horizontal direction the velocity of the ball bearing is dependent on the same time:
v=s/t
v = 0.335m/0.136s = 2.46m/s

So that seems to be the velocity of the ball bearing in the horizontal direction, given that rolling friction should be minimal.

to calculate how far the ball bearing should travel in the horizontal direction I have to work out the time it takes to fall.

so velocity in a vertical direction of the ball travelling down the paper tube is:
v^2 = u^2 +2as
v^2 = 0 + 2 x 9.8 x 0.09
v = 1.33m/s

this becomes the initial velocity of the ball when it falls off the edge of the table. the time to fall from the table comes from:
s = ut + 1/2 at^2

0.915m = 1.33 x t + 1/2 x 9.8 x t^2
0 = 4.9t^2 + 1.33t - 0.915

which I then solve using the quadratic equation formula

t = [-1.33 +/- (1.33^2 -4x4.9x-0.915)^1/2]/2x4.9

t = 0.32s

so if the horizontal velocity = 2.46m/s and it falls for 0.32s, it stands that the distance it travels in the horizontal direction until it strikes the floor is:
s=vt
s= 2.46m/s x 0.32s
= 0.76m

so that seems right, but here is my problem:
when I physically measure the distance it travels in the horizontal direction to the floor, I measure 0.46m. this seems a fairly large difference to me, and working backwards suggests that the horizontal velocity is actually 1.43m/s.

I would like to know what element I am missing. I thought initially it was friction or air drag, but I know that friction from the paper is small. can anyone tell me why my calculation is wrong compared to the physical measurement?

any help would be greatly appreciated!