Mechanics | Relativity Answers

A particle starts from rest and moves in a straight line on a smooth horizontal surface. Its acceleration at time t seconds is k(4v + 1)m/s^2 , where k is a positive constant and v m/s is the speed of the particle. Given that v = (e^2 - 1)/4 when t = 1, show that v = 1/4(e^(2t) - 1)

A smooth sphere A of mass 3m moving on a smooth horizontal table with speed 4u, impiges directly on another smooth sphere B of mass 2m, moving with speed u in the opposite direction to A. The coefficient of restitution between A and B is e.
(a) Find the impulse exerted on A by the impact.
At the moment of the impact, the line of centres of the spheres is perpendicular to a vertical wall which is at a distance x from the point of collision and nearer to B than to A, and B subsequently collides with the wall.
(b) Find in terms of x, the distance of A from the wall at the instant B hits the wall.

A smooth sphere A of mass 3m moving on a smooth horizontal table with speed 4u, impiges directly on another smooth sphere B of mass 2m, moving with speed u in the opposite direction to A. The coefficient of restitution between A and B is e.
(a) Find in terms of e and u, the speed of B after the impact.
(b) Show that e = 1/3
(c) Find the impulse exerted on A by the impact.

A uniform ladder of weight W abd length 2a rests in limiting equilibrium with one end on a rough horizontal ground and the other end on a rough vertical wall. The coefficients of friction between the ladder and the ground and between the ladder and the wall are u and t respectively.
(a) Show that 5u + 6tu - 6 = 0
(b) Find the values of u and t, given that ut = 1/2
(c) Show by integration that tge centroid of a uniform semi-circular lamina of radius a from the centre is 4a/(3*pie).

A tyre of mass 15kg is suspended on a rope of length 7.5m and a girl of mass 75kg sits in the tyre. If the girls speed at the lowest point as she swings is 5m/s and the rope extends by 20cm, find the radius of this rope(young modulus = 2.0 x 10^11 Pa)

A body of mass m wants to leave completely from the earths gravitational attraction, find the minimum speed it requires at the earth surface (escape velocity)

Consider a planet of mass M with an orbiting satellite at a distance of R from each of their centers. Show that Kepler's third law of gravitation is true.

An oscillatory motion has a period of 50.0s and an amplitude of 20.0cm. Determine its maximum acceleration and mechanical energy

The speed of light C is related to the permeability and thr permittivity e by the expression
C^2*uo*e = 1
(i) Show that this equation is homogenous
(ii) Calculate the value of uo