Question

1. Derive the expressions for elastic collision in one dimension and also discuss their two special cases?2. Derive the equations of a uniformly accelerated motion by calculus method.

Answer 1

2- Derivation of equations by calculus method:
$$\begin{gathered} 1-\mathrm{We}\mathrm{know}\mathrm{that}, \\ \mathrm{Acceleration},\mathrm{a}=\frac{\mathrm{dv}}{\mathrm{dt}} \\ \mathrm{or}{\int }_{\mathrm{u}}^{\mathrm{v}}1d\mathrm{v}={\int }_0^{\mathrm{a}}1d\mathrm{t} \\ \mathrm{or}\mathrm{v}-\mathrm{u}=\mathrm{at} \\ \mathrm{or}\mathbf{v}\mathbf{=}\mathbf{u}\mathbf{+}\mathbf{at} \\ 2- \\ \mathrm{a}=\frac{\mathrm{vdv}}{\mathrm{ds}} \\ \mathrm{or}{\int }_{\mathrm{u}}^{\mathrm{v}}\mathrm{v}d\mathrm{v}={\int }_0^{\mathrm{s}}\mathrm{a}d\mathrm{s} \\ \mathrm{or}(\frac{{\mathrm{v}}^2}{2}{{)}_{\mathrm{u}}}^{\mathrm{v}}=\mathrm{as} \\ \mathrm{or}{\mathbf{v}}^{\mathbf{2}}\mathbf{-}{\mathbf{u}}^{\mathbf{2}}\mathbf{=}\mathbf{2}\mathbf{as} \\ 3- \\ \mathrm{Putting}\mathrm{v}=\mathrm{u}+\mathrm{at}\mathrm{in}\mathrm{equation}{\mathrm{v}}^2-{\mathrm{u}}^2=2\mathrm{as}, \\ (\mathrm{u}+\mathrm{at}{)}^2-{\mathrm{u}}^2=2\mathrm{as} \\ \mathrm{or}2\mathrm{uat}+{\mathrm{a}}^2{\mathrm{t}}^2=2\mathrm{as} \\ \mathrm{or}\mathbf{s}\mathbf{=}\mathbf{ut}\mathbf{+}\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{at}}^{\mathbf{2}} \end{gathered}$$