Kinematics - projectile motion of a falling body

[Jo Turner]

Homework Statement

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 traveling 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!

 

[jbsteele]

Homework Equations x = ut + 1/2at^2v^2 = u^2 +2asv=s/tThe Attempt at a SolutionIt appears that you are making a few assumptions in your calculations that may be leading to the discrepancy between your calculated distance and the measured distance. Firstly, you are assuming that the initial velocity of the ball bearing is 0 when it starts rolling down the paper tube. However, this is not necessarily true, as the ball bearing may have had some momentum before it started rolling down the tube. This could cause the ball bearing to travel further than expected. Additionally, you are assuming that there is no air resistance or friction in the system. While the friction from the paper tube may be minimal, the air resistance could be significant. This could also cause the ball bearing to travel further than expected. Finally, you have assumed that the acceleration due to gravity is 9.8 m/s^2. However, this is only a rough approximation and the actual value could be slightly different from this value. This could also lead to a discrepancy in your calculated results. In order to get a more accurate result, you may need to measure the initial velocity of the ball bearing before it starts rolling down the paper tube and take into account the air resistance and other factors that may affect the motion of the ball bearing.

 

[kerimek]

I would first like to commend you on your thorough approach to solving this problem. Your use of the kinematic equations and careful measurements are all important steps in the scientific process. However, I believe there may be a flaw in your calculation that is causing the discrepancy between your predicted and measured values.

In your calculation for the time it takes for the ball bearing to fall from the table, you use the equation s = ut + 1/2 at^2. However, this equation is only valid when the initial velocity (u) is 0. In this case, the initial velocity is actually 1.33m/s, as you calculated earlier. Therefore, the correct equation to use would be v = u + at, which would give a different value for the time it takes for the ball bearing to fall.

Using this equation, we get:

1.33m/s = 9.8m/s^2 x t
t = 0.136s

This value is consistent with the time you calculated earlier for the ball bearing to roll down the paper tube. Using this value for time, your calculation for the horizontal distance traveled by the ball bearing should give a value closer to your measured distance of 0.46m.

I hope this helps to resolve your issue. Keep up the good work in your scientific explorations!