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titansarus

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## Homework Statement

We shot a projectile with mass ##m## and velocity ##v_0## with angle ##\phi## it collide with a box with mass ##M## at the maximum height of its path. Then, they both start to move with another speed. (We define ##t=0## at this time) (Completely Inelastic Collision). The box is also connected to a spring from the ceiling with spring constant ##k## and the table has kinetic friction coefficient ##\mu_k##. At the time ##t=t_0## it stops for a moment and then move backward (because of spring force). We want to find these:

(The angle of spring will be small so ##sin\theta = tan\theta = \theta## and ##cos \theta = 1 - \theta^2 /2##

I) Work done by spring from from ##0## to ##t_0##.

II) Work done by friction from from ##0## to ##t_0##.

III) FInd an equation to find the position at time ##t_0##. (Solving it is optional)

## Homework Equations

##f = -k x##

##dw = \vec f . d\vec r##

##f = \mu_k N##

##sin\theta = tan\theta = \theta## and ##cos \theta = 1 - \theta^2 /2##

## The Attempt at a Solution

I can find the velocity at t=0: ##m v_0 cos \phi = (m + M) v## so ##v = v_0 cos\phi m/(m+M)##.

I can also say that ##f_{spring} = -k\Delta x = k (L - L / (1 - \theta^2 /2)) = -k L (\theta^2 /2) / (1- \theta^2 /2)##

and ##f.dr = f.dx = - k L (\theta^2 /2) /(1- \theta^2 /2) sin \theta L d\theta = - k L^2 (\theta^3 /2) /(1- \theta^2 /2) d\theta##

But I cannot find where velocity gets zero (##t = t_1##) to find the ##\theta## needed for integration.

Maybe I am getting the problem more complicated that it is? I don't know, maybe no integration is needed because ##\theta## is small enough that we can say it is equal to ##d\theta##.

So what do I do?

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