Conservation of Energy Question

Rock00

Question is:
------------

Two children are playing a game in which they try to hit a small box on the floor with a marble fired from a spring loaded gun that is mounted on a table. The target box is 2.20 m horizontallyl from the edge of the table. Bobby compresses the spring 1.10 cm, but the marble falls 27.0 cm short. How far shuld Rhoda compress the spring to score a hit?

Answer:
--------

Diagram:

_______ projectile |
_|_spring_|-------------___ v
| | marble<--- ---O_
| | --__
| | --__
| |__________________________|box|_
|<------------2.20 m ---------->|

Given
-----
dx = 2.20 m
Sc (spring compressed) = .011 m
(falls .27 m short)

Sc=?
(hits the targe)

so basically what should be the compression of the spring before it can hit the target?

-----------------
I didn't know how to start. I wrote down F=-ks. I figured that the initial velocity is 0. PE at start is 0. Total change in energy is 0. But I do not know what else to do. Please help!!!

Rock00

Attached is a clearer diagram of the previous posted diagram!

Attached Thumbnails

 

LeonhardEuler

Are you sure that the PE at the start is 0? What about the compression in the spring or the hieght of the object?

Homer Simpson

with kinematics you should be able to find the horizontal velocity that got the projectile to go 220 - 27 cm. Knowing that velocity should let you solve for the spring constant. With this knowledge, find out what horizontal velocity will get the projectile to go where you want it to go. Then using spring equations again, figure out how much spring compression will make that initial velocity.

Does this seem to help?

LeonhardEuler

However, since he does not know the hieght of the table, he will not be able to find the spring constant. It doesn't matter though. Solve for the height in terms of the spring constant and both terms will cancel out in the end.

marlon

Ahh, this is a classic one.

You will be needing :
1) split up the problem in two parts (one part with the spring on tha table, the second part when the object has left the table and fall down because of gravity)

2) Nexton's second law (determin all the forces in the horizontal and vertical direction when the object is falling down : gravity)

3) conservation of total energy when the particle just leaves the table. This will be the "connection" between the particle just leaving the spring (calculate the energy it gets from the spring) and the motion under gravity.

good luck

marlon

edit : Both k and m will not be important for your calculation. You do not need to know them, because they will cancel out

Homer Simpson

Oh yeah, I missed that detail. That makes it much more tricky. Build a model.

marlon

Guys, just use k and m for springconstant and mass. You will see that both quantities will cancel out (if you solve this question correctly :) ). You don't need them...

marlon

Rock00

Hi, do you think you can show me the process of how you started the problem. I would greatly appreciate it.

Thanks

LeonhardEuler

The way I did the problem was to write equations for the x- and y-coordinates as functions of time. Then I solved for time in the x equation, and plugged the expression I got for t into the y equation. This gives y as a function of x. The expression for x has an initial velocity, v_0 in it. Use conservation of energy to solve for v_0 in terms of k, the spring constant, m, the mass, and s, the displacement of the spring. Then it is just a matter of using the information given to solve for h in terms of k, m, and s. k and m cancel away in the end.

marlon

First forget about the 1.73 cm. That's the solution of another question. Sorry...

well, start with conservation of energy : [tex]\frac{kx^2}{2} = \frac{mv^2}{2}[/tex] ; x is the distance the spring is compressed.

From this, you get a velocity (just bring on k and m...don't mind them) when x = 0.011m

When leaving the table, the object moves under the influence of gravity.
Applying Newton's second law gives you

[tex]y = \frac{gt^2}{2}[/tex]
[tex]x' = vt[/tex] ; x' is the horizontal distance travelled by the object. You know that x' = 1.93 m if the spring is compressed 0.011 m.

From the above two equations, write y as a function of x' and v. Calculate the y-value with the 0.011m and 1.93m data.

Then, using this y-value (k and m are still not known) you can get the x-value (distance that the spring is compressed) when you know that x' = 2.2m

marlon

marlon

Correctamundo

marlon

Rock00

Where are you getting the 2 from and what is the t for?

is 2 part of the equation and t is for time and if t is time then how would you get time or 1.93 m.

Thank you

Rock00

does anyone know how marlon got [tex]y = \frac{gt^2}{2}[/tex]

any help appreciated

marlon

Don't you know the equation for the vertical position of an object moving under the influence of gravity ?

In general, the formula is :

[tex]y = y_0 + v_{0y}t + \frac{a_y t^2}{2}[/tex]

[tex]y_0 [/tex] is the initial position ; for you it is 0

[tex]v_{0y}[/tex] is the initial velocity component in the y direction ; for you this is 0 because the v after energy conservation only has a component along the x direction.

[tex]a_y[/tex] is the acceleration in the y direction ; for gravity this is g


regards
marlon

Rock00

Thank you marlon. I still do not know what to do after I get the Y.

I got Y=301686.116/(k*m)

Help would be appreciated.

marlon

read my first post carefully. it's all in there

marlon