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### 2-D Collision

A ball of mass 10 kg moving with a velocity $10\sqrt 3$ m/s along x-axis, hits another ball of mass 20 kg which is at rest kept at the origin. After collision, the first ball comes to rest and the second one disintegrates into two pieces. One of the pieces moves along y-axis at a speed of 10m/s. The second piece of mass m (say) moves at a speed of 20m/s at an angle $\theta$ (degree) with respect to the x–axis. Find the values of m and $\theta$.

Solution

$10\times 10 \sqrt 3 + 20 \times 0 = 10 \times 0 + m \times 20 \cos \theta + (20-m) \times 0$

$\Rightarrow 5\sqrt 3 = m \cos \theta$ ........(A)

Now, COLM along y direction,

$10 \times 0 + 20 \times 0 = 10 \times 0 - m \times 20 \sin \theta + (20-m) \times 10$

$\Rightarrow 2m \sin \theta = 20 - m$ ........(B)

From (A) & (B),

$4m^2 = (20-m)^2 + 300$

$\Rightarrow 3m^2 + 40m - 700 = 0$

$\Rightarrow m = 10 kg$ rejecting the negative value. This also means that the two pieces will have the same mass.

From (A), $\cos \theta = \frac {\sqrt 3}{2}$

$\Rightarrow \theta = 30^\circ$

### Sum of the coefficients in the expansion of $(x+y)^n$ ....

If the sum of the coefficients in the expansion of $(x+y)^n$ is 4096, then the greatest coefficient in the expansion is _ _ _ _ . Solution $C_0 + C_1 + C_2 + C_3 + ......................... + C_n =4096$ $\therefore 2^n = 4096 =2^{12}$ $\Rightarrow n = 12$ Greatest coefficient = ${}^{12}{C_6} = 924$