Canning Theory (projectile motion, error propigation & perfectly elastic collisions)

In summary, the conversation discusses how to place a can on a table so that a ball rolling off the end of the table will land in the can. Various expressions are found for the distance from the table top at which the can should be placed, the time it takes for the ball to reach the top of the can, the speed of the ball when it reaches the top of the can, the maximum fractional error in the measurement of the initial speed of the ball, the maximum vertical deviation of the velocity vector as the ball leaves the table top, and the number of collisions the ball will make with the wall of the can before hitting the bottom. Calculations are shown for some of these expressions, while assistance is needed for others.
  • #1
2011vlu
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Homework Statement


A ball rolls off the end of a table of height H. The ball has speed S when it leaves the horizontal table top. We wish to place a can of height h and diameter D at the correct distance from the table’s edge so that the ball lands in the can (aim for the top center of the can!) Find expressions for the following in terms of any/all of H, h, D, S, and g.

a) The distance from the table top (x) at which we should place the center of the can.
b) The time it will take the ball to reach the top of the can.
c) The speed the ball will be moving when it reaches the top of the can.
d) The maximum fractional error in our measurement of S which is allowed so that we will still get the ball in the can (assuming perfectly horizontal launch from the table top).
e) The maximum vertical deviation (|[tex]\delta[/tex]Vy |) for the velocity vector as the ball leaves the table top that is allowed if we are to still get the ball in the can. Assume that the speed of the ball as it leaves the table top is still S.
f) Assuming a perfect insertion into the middle of the top of the can and that the effect of a collision with the walls of the can is "perfectly elastic"; meaning that the effect of a collision with the wall of the can is to instantaneously reverse the horizontal component of the ball's velocity, develop an expression for the number of collisions the ball will make with the wall of the can before it hits the bottom of the can (where there is some sticky stuff which stops the ball instantly).

Homework Equations


x(t)=x0+(v0cos[tex]\theta[/tex])t
y(t)=y0+(v0sin[tex]\theta[/tex])t-(1/2)gt^2

The Attempt at a Solution


a) I solved for t when y(t) is h because that's how tall the can is. I got x(t)=S(rad((2H-2h)/g))
b) I already solved for t in the above section, and I got t=rad((2H-2h)/g)
c) I found the derivative of the position to get the velocity, found the magnitude of that vector, and got rad(S^2-2gH-2gh)
d) I'm stuck, I THINK what I'm supposed to do is S(rad((2H-2h)/g))-D=x(rad((2H-2h)/g)), but it doesn't quite work.
e) Please help with this part.
f) Please help with this part.
 
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  • #2


c) I found the derivative of the position to get the velocity, found the magnitude of that vector, and got rad(S^2-2gH-2gh)
v = Vi - g*t = -g*t (no S involved in the vertical motion!)

For (d) I would start with the extra speed needed to hit the far edge of the can divided by the center speed.

(delta S)/S = (x + D/2)/t - x/t all divided by x/t
it works out very simply to .5*D/x as it should since everything is linear.

I played with (e) a bit. I used Sx and Sy for the speed components and worked the problem again with horizontal distance x + D/2. Got an expression for Sy (which is the same as delta Sy) that has Sx in it a couple of times. Could get rid of that using Sx^2 + Sy^2 = s^2.
Pretty messy. I'm sure you have to work the problem with an initial vertical velocity but perhaps there is a way to avoid the mess assuming small deviations and using calculus.
 
  • #3


a) The distance from the table top at which we should place the center of the can can be found by setting the y-position equal to the height of the can (h) and solving for x:
x = (S^2/g)*sin(2θ) - (2h/g)*cos^2(θ)

b) The time it will take the ball to reach the top of the can can be found by setting the y-position equal to the height of the can (h) and solving for t:
t = (S/g)*sin(θ) + sqrt((S^2/g^2)*sin^2(θ) + (2h/g))

c) The speed of the ball when it reaches the top of the can can be found by taking the magnitude of the velocity vector at the time t found in part b:
v = sqrt(S^2 + 2gh)

d) The maximum fractional error in the measurement of S can be found by setting the distance from the table top (x) equal to the distance from the table top (x) plus the diameter of the can (D) and solving for S:
S = (g/2h)*(D + sqrt(D^2 + 4gh))

e) The maximum vertical deviation in the velocity vector can be found by considering the maximum possible error in the angle θ, which would result in the maximum possible deviation in the vertical component of the velocity. This can be found by taking the derivative of the vertical position equation with respect to θ and setting it equal to 0:
d/dθ (y(t)) = (S/g)*cos(θ) - (2h/g)*sin(θ) = 0
Solving for θ:
θ = tan^-1(S/(2h))

The maximum vertical deviation in the velocity vector is then given by:
|δVy| = S*sin(θ) = S*sin(tan^-1(S/(2h)))

f) The number of collisions the ball will make with the wall of the can before hitting the bottom can be found by considering the distance the ball travels horizontally before each collision and dividing that by the diameter of the can (D). This will give the number of collisions, n:
n = x/D = (S^2/g)*sin(2θ) / D

Note: These equations assume that the ball is launched horizontally from the table top and that the can is placed directly under the
 

1. How is projectile motion related to canning theory?

Projectile motion is a fundamental concept in canning theory. The motion of a can being launched from a canning machine can be described as a projectile, with a parabolic trajectory due to the influence of gravity.

2. What is error propagation and why is it important in canning theory?

Error propagation is the process of quantifying how uncertainties in measured parameters affect the final result of a calculation. In canning theory, it is important to consider error propagation when determining the accuracy of the canning process and the quality of the final product.

3. How do perfectly elastic collisions play a role in canning theory?

In canning theory, perfectly elastic collisions refer to the ideal scenario where there is no loss of energy during a collision between two objects. This concept is relevant in canning as it helps to determine the forces and velocities involved in the canning process, and can also impact the shelf life and safety of the canned product.

4. What factors can affect the accuracy of canning theory?

There are several factors that can affect the accuracy of canning theory, including the precision of measuring equipment, the consistency of the canning process, and external factors such as temperature and atmospheric pressure. It is important to account for these factors when conducting canning experiments and making calculations.

5. How can canning theory be applied in real-world situations?

Canning theory has many practical applications, particularly in the food industry. It can be used to optimize the efficiency and safety of canning processes, as well as to ensure the quality and shelf life of canned products. Additionally, canning theory can also be applied in other fields such as physics, engineering, and materials science.

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