|Exercise 2 A particle of constant mass is subject to to a force . Suppose that for e the velocities are and , respectively. Show that the work the force does on the particle equals its change in kinetic energy.
|Exercise 3 For the conditions of exercise 1 calculate:
|Exercise 4 Show that if a particle is subject to and is its instantaneous velocity then the instantaneous power is
By definition it is . Hence
|Exercise 5 Show that the integral is independent of a particle’s trajectory if and only if .
Let denote a closed curve and admit that is path independent. Then
Where the last equality follows from our assumption that is path independent.
Suppose now that .
Where the last equality follows from our . Hence .
|Exercise 6 A particle of mass moves along subject to to a conservative force field . If the particle’s positions are and on and , respectively, show that, if is the total energy then
Write solve in order to and integrate.
|Exercise 7 Consider a particle of mass that moves vertically on a resistive medium where the retarding force is proportional to the particle’s velocity. Consider that particle initially moves in the downward direction with an initial velocity from an height . Derive the particle’s equation of motion .
It is with and . Then solve in order to and integrate to find (the terminal velocity is ).
Solving in order to and integrating and it is .
|Exercise 8 A particle is shot vertically on a region where exists a constant gravitational field with a constant initial velocity . Show that, in the presence of resistive force proportional to the square of the particle’s instantaneous velocity, the velocity of the particle when it returns to its initial position is:
Where denotes the particle’s terminal velocity.