
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
[asy]size(8cm);

real w = 2.718; // width of block
real W = 13.37; // width of the floor
real h = 1.414; // height of block
real H = 7; // height of block + string
real t = 60; // measure of theta

pair apex = (w/2, H); // point where the strings meet

path block = (0,0)--(w,0)--(w,h)--(0,h)--cycle; // construct the block
draw(shift(-W/2,0)*block); // draws white block
path arrow = (w,h/2)--(w+W/8,h/2); // path of the arrow
draw(shift(-W/2,0)*arrow, EndArrow); // draw the arrow

picture pendulum; // making a pendulum...
draw(pendulum, block); // block
fill(pendulum, block, grey); // shades block
draw(pendulum, (w/2,h)--apex); // adds in string

add(pendulum); // adds in block + string
add(rotate(t, apex) * pendulum); // adds in rotated block + string

dot(" $\theta$ ", apex, dir(-90+t/2)*3.14); // marks the apex and labels it with theta
draw((apex-(w,0))--(apex+(w,0))); // ceiling
draw((-W/2-w/2,0)--(w+W/2,0)); // floor[/asy]
A block of mass  $m=\text{4.2 kg}$  slides through a frictionless table with speed  $v$  and collides with a block of identical mass  $m$ , initially at rest, that hangs on a pendulum as shown above. The collision is perfectly elastic and the pendulum block swings up to an angle  $\theta=12^\circ$ , as labeled in the diagram. It takes a time  $ t = \text {1.0 s} $  for the block to swing up to this peak. Find  $10v$ , in  $\text{m/s}$  and round to the nearest integer. Do not approximate  $ \theta \approx 0 $ ; however, assume  $\theta$  is small enough as to use the small-angle approximation for the period of the pendulum. 

*(Ahaan Rungta, 6 points)*

Present the answer in LaTex format: \boxed{Your answer}
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