Question

Find the net work W done on the particle by the external forces during the motion of the particle in terms of the initial and final kinetic energies.Express your answer in terms of Kinitial and Kfinal.W=

Answer 1

Answer:

$W=K_f-K_i$

Explanation:

The work done on a particle by external forces is defined as:

$W=\int\limits^{r_f}_{r_i} {F\cdot dr} \,$

According to Newton's second law $F=ma$. Thus:

$W=\int\limits^{r_f}_{r_i}{ma\cdot dr} \,$

Acceleration is defined as the derivative of the speed with respect to time:

$W=m\int\limits^{r_f}_{r_i}{\frac{dv}{dt}\cdot dr} \,\\\\W=m\int\limits^{r_f}_{r_i}{dv \cdot \frac{dr}{dt}} \,$

Speed is defined as the derivative of the position with respect to time:

$W=m\int\limits^{v_f}_{v_i} v \cdot dv \,$

Kinetic energy is defined as $K=\frac{mv^2}{2}$:

$W=m\frac{v_f^2}{2}-m\frac{v_i^2}{2}\\W=K_f-K_i$