A trolley of mass $200 \mathrm{kg}$ moves with a uniform speed of $36 \mathrm{km} {\mathrm{h}}^{-1}$ on a frictionless track. A child of mass $20 \mathrm{kg}$ runs on the trolley from one end to the other ( $10 \mathrm{m}$ away) with a speed of $4 \mathrm{m} {\mathrm{s}}^{-1}$ relative to the trolley in a direction opposite to the its motion and jumps out of the trolley. What is the final speed of the trolley? How much has the trolley moved from the time the child begins to run?

(i) Dot product or scalar product of two vectors A and B is defined as $A.B= A\times B\times \cos \theta$, where θ is the angle between A and B

(ii) $A= A_x\hat{i}+A_y\hat{j}+A_z\hat{k}$ and $B= B_x\hat{i}+B_y\hat{j}+B_z\hat{k}$ then, $A.B= A_x{B}_x+A_y{B}_y+A_z{B}_z$

(iii) Work done by a constant force, $W= \overrightarrow{F}.\overrightarrow{s}$

(iv) Work done by a variable force $dW= \overrightarrow{F}.d\overrightarrow{s}$

2. Relation between kinetic energy and momentum: 

Relation between momentum and kinetic energy, $KE= \frac{P^2}{2m}$ and $P= \sqrt{2m\times KE}$, where $P$ is linear momentum

3. Potential energy:

(i) Potential energy $\left(U\right)$ and force are related as, ${\int }_{U_1}^{U_2}dU= -{\int }_{r_1}^{r_2}\overrightarrow{F}.d\overrightarrow{r}$ i.e., $U_2-U_1= -{\int }_{r_1}^{r_2}\overrightarrow{F}.d\overrightarrow{r}=-W$ where $W$ is the work done by the conservative force.

(ii) Conservative force can be found from potential energy using the equation, $F\left(x\right)= -\frac{dU}{dx}$

4. Work energy theorem:

According to work-energy theorem, the net work done on a body is equal to the change in the kinetic energy, $W_{net}= \Delta KE$

5. Power:

(i) Power is the rate of work done, $P= \frac{dW}{dt}$

(ii) The average power delivered by an agent is given by $P_{av}= \frac{W}{t}$

(iii) Instantaneous power, $P= \frac{FdS}{dt}= F\frac{dS}{dt}= F.V$

6. Conservation of energy:

(i) According to conservation of mechanical energy, the total mechanical energy of an isolated system is conserved if the internal forces are conservative.

7. Collisions:

(i) The total momentum of a system remains constant during a collision.

(ii) Gravitational force and spring force are always non-impulsive during collision.

(iii) Total energy is conserved in an elastic collision.

(iv) The energy lost is maximum in a perfectly inelastic collision.

(v) The two equations which we can use in a collision are: $m_1{u}_1+m_2{u}_2= m_1{v}_1+m_2{v}_2$ and $e(u_1-u_2)= v_2-v_1$.

(vi) $e$ is the coefficient of restitution. It is $1$ for elastic collision, $0$ for perfectly inelastic collision and for inelastic collision it will be between $0$ and $1$.