Vector in Mechanics Assignment

)The position vector of a particle at time t seconds is given by r= (1-sin t) i-cost j. Show that the acceleration of the particle is always perpendicular to its velocity. (4 Marks)
b)A particle of mass 0.2 Kg is moving along a course of 030° at 25m/s. The particle is acted upon by a force of 15N for 0.4 seconds so that the subsequent direction of motion is 095°. Determine
The direction of the force
The final velocity of the particle
Question Two
a)A particle of mass m moves in the Cartesian plane so that its position vector around an elliptic path is given as r=a sin ß t i +b cos ßt j, where a, b and ß are positive scalars and a>b.
Show that the force field is conservative
Determine the potential at the points A (a, 0) and B (0, b).
Determine the work done by the force in moving the particle from A (a, 0) and B (0, b).
Show that the total energy of the particle is a constant. (10 marks)
Question Three
a)Show that the force field F= (y2z3-6xz2) i + 2xyz3j + (3xy2z2-6x2z) k is conservative and hence determine the scalar potential f(x, y, z) such that F=???.
(5 marks)
b)If the acceleration of a particle moving in a straight line with simple harmonic motion??x?^2 when the displacement from a central position is x, prove that the vewlocity V opf the particle is given by v^2= ?^2(a^2-x^2) where a is the amplitude. (5 marks)