Work, Energy and Power

Using conversation of momentum in direction perpendicular to the original direction of motion,

mv₁ sin 30° = mv₂ sin 30°

$\Rightarrow \frac{v_1}{v_2}=1$

Power calculations can also differ based on the nature of forces, which are as follows:

• Conservative Forces: Here, work done is path-independent. e.g., gravity, spring force
• Non-Conservative Forces: Work depends on the path. e.g., friction

by conservation of mechanical energy

              K.E_i + P.E_i = K.E_f + P.E_f

              $\frac{1}{2}*0.5v^2+0=\frac{1}{2}*0.5{\left(\frac{v}{2}\right)}^2+\frac{1}{2}k{x}^2$  

              $\Rightarrow \frac{1}{2}*0.5\left[v^2-\frac{v^2}{4}\right]=\frac{1}{2}k{x}^2$  

              $\Rightarrow \frac{1.5}{4}12*12=\frac{k*9}{100}$  

              $\frac{1.5*36}{9}*100=k$  

              k = 600 N/m¹

P = F.v = 5+3-6 = 2W

Velocity just before striking the ground
v? = √2gh
v? = √ (2*10*10) = 10√2 m/s
v? = -10√2?
If it reaches the same height, speed remains same after collision only the direction changes.
v? = 10√2 m/s
v? = 10√2?
|Impulse| = m|Δv|
= m|10√2? - (-10√2? )|
= 0.15 [2 (10√2)]
= 3√2 kg m/s
= 4.2 kg m/s

$\frac{1}{2}\mathrm{m}{\left(\frac{\mathrm{u}}{3}\right)}^2-\frac{1}{2}{\mathrm{m}\mathrm{u}}^2=-{\mathrm{F}}_{\mathrm{R}}*24$

$0-\frac{1}{2}{\mathrm{m}\mathrm{u}}^2=-{\mathrm{F}}_{\mathrm{R}}*\mathrm{d}$

$\frac{\frac{1}{2}{\mathrm{m}\mathrm{u}}^2}{\frac{1}{2}m{u}^2*\frac{8}{9}}=\frac{\mathrm{d}}{24}$

$\mathrm{d}=24*\frac{9}{8}=27\mathrm{c}\mathrm{m}$

Potential energy stored in the spring $=\frac{1}{2}{\mathrm{k}\mathrm{x}}^2$

$\begin{array}{rr}& \text{Now}\frac{1}{2}\mathrm{k}(2{)}^2=\mathrm{U}\\ & \&\frac{1}{2}\mathrm{k}(8{)}^2=U^{\mathrm{\text{'}}}\mathrm{}\text{(say)}\\ & \Rightarrow U^{\mathrm{\text{'}}}=\frac{64}{4}U=16U\end{array}$

${\mathrm{P}}_{\text{in}}=\frac{\mathrm{m}\mathrm{g}\mathrm{h}}{\mathrm{t}}=\frac{15*10*60}{1}=9000\mathrm{w}$

${\mathrm{P}}_{\text{out}}=90\mathrm{\%}$ of ${\mathrm{P}}_{\text{in}}$

$\Rightarrow 8.1\mathrm{k}\mathrm{w}$

$\mathrm{U}+\mathrm{K}\mathrm{E}=\mathrm{E}$

$4\mathrm{U}=\mathrm{E}=\mathrm{m}\mathrm{g}\mathrm{S}$

$4\mathrm{m}\mathrm{g}\mathrm{h}=\mathrm{m}\mathrm{g}\mathrm{S}$

$\mathrm{h}=\frac{\mathrm{S}}{4}$

$\mathrm{v}=\sqrt[]{2\mathrm{g}\left(\frac{3\mathrm{S}}{4}\right)}=\sqrt[]{\frac{3\mathrm{g}\mathrm{S}}{2}}$

$i_{\text{series}}=\frac{E}{R_{\text{series}}}$

$\Rightarrow {\mathrm{i}}_{\text{series}}=\frac{\mathrm{E}}{10\mathrm{R}}$

${\mathrm{i}}_{\text{parallel}}=\frac{\mathrm{E}}{{\mathrm{R}}_{\text{Parallel}}}$

$=\frac{\mathrm{E}}{\mathrm{R}/10}$ ${\mathrm{i}}_{\text{parallel}}=\mathrm{n}*{\mathrm{i}}_{\text{series}}$

$\Rightarrow \frac{10\mathrm{E}}{\mathrm{R}}=\frac{\mathrm{n}\mathrm{E}}{10\mathrm{R}}$

$\Rightarrow \mathrm{n}=100$