As shown in the figure, a particle of mass 10 kg is placed at a point A. When the particle is slightly displaced to its right, it starts moving and reaches the point B. The speed of the particle at B is x m/s. (Take g = 10 m/s²) The value of 'x' to the nearest integer is----------.

Vertical component of velocity just after collision $=\frac{u}{2\sqrt[]{2}}$

$k_i=\frac{m{u}^2}{2}{k}_f=\frac{1}{2}m\frac{u^2}{2}+\frac{1}{2}m\frac{u^2}{8}=\frac{5m{u}^2}{16}$ $\frac{}{\text{exception 25:}}$

Fraction  $=\frac{k_i-k_f}{k_i}=1-\frac{\frac{5m{u}^2}{16}}{\frac{m{u}^2}{2}}=1-\frac{5}{8}=\frac{3}{8}$

For elastic collision   ${\mathrm{K}\mathrm{E}}_{\mathrm{i}}={\mathrm{K}\mathrm{E}}_{\mathrm{r}}$

$\frac{1}{2}\mathrm{m}\times 25+\frac{1}{2}\times \mathrm{m}\times 9=\frac{1}{2}\mathrm{m}\times 32+\frac{1}{2}m{v}^2$

$34=32+v^2$

$\mathrm{K}\mathrm{E}=\frac{1}{2}\times 0.1\times 2=0.1\mathrm{J}=\frac{1}{10}$

$x=1$

No. Since kinetic energy is a scalar quantity, it only depends on speed of the body and not the direction. So if the direction of the body is changed but the speed remains unchanged, there won't be any effect on the kinetic energy. However, if changing the direction also changes the speed of the body, then kinetic energy of the body will also change.

If you look closely at the formula of kinetic energy (1/2*m*v^2), the velocity is squared which automatically gives a positive integer. And mass of the body can never be a negative value, which leads to the result being a positive integer.

The 1/2 is a result of mathematical calculation, which occurs when we integrate? vdv in the formula of work done according to Newton's second law of motion. Without this, the final result will turn out to be twice of the actual value.