A particle of mass `m` moves in pane `xy` due to the force varying with velocity as `F=a(dot(y)-dot(x)j)`, where `a` is a positve constant, `i` and `j` are the unit vectors of the `x` and `y` axes. At the initial moment` t=0` the particle was located at the point `x=y=0` and possessed a velocity `v_(0)` directed along the unit vector `j`. Find the law of motino `x(t), y(t)` of the particle, and also the equation of its trajectory.

Find the potential varphi (x,y) of an electrostatic field E = 2axyi + a (x^(4) - y^(2)) j , where a is a constant, I and J are the unit vectors of the x and y axes.

A particle moves in the plane xy with velocity v=ai+bxj , where i and j are the unit vectors of the x and y axes, and a and b are constants. At the initial moment of time the particle was located at the point x=y=0 . Find: (a) the equation of the particle's trajectory y(x) , (b) the curvature radius of trajectory as a function of x.

The velocity of a particle moving in the positive direction of the x axis varies as v=5sqrtx , assuming that at the moment t=0 the particle was located at the point x=0 , find acceleration of the particle.

A particle of mass m moves due to the force F=-alpha m r, where alpha is a positve constant , r is the radius vector of the particle relative to the origin of coordinates. Find the trajectory of its motion if at the initial moment r=r_(0)i and the velocity v=upsilon_(0)j , whee i and j are the unit vectors of the x and y axes.

A particle projected from origin moves in x-y plane with a velocity vecv=3hati+6xhatj , where hati" and "hatj are the unit vectors along x and y axis. Find the equation of path followed by the particle :-

A particle projected from origin moves in x-y plane with a velocity vecv=3hati+6xhatj , where hati" and "hatj are the unit vectors along x and y axis. Find the equation of path followed by the particle :-

A particle projected from origin moves in x-y plane with a velocity vecv=3hati+6xhatj , where hati" and "hatj are the unit vectors along x and y axis. Find the equation of path followed by the particle :-

The velocity of a particle moving in the x direction varies as V = alpha sqrt(x) where alpha is a constant. Assuming that at the moment t = 0 the particle was located at the point x = 0 . Find the acceleration.

A radius vector of a point A relative to the origin varies with time t as r=ati-bt^2j , where a and b are positive constants, and I and j are the unit vectors of the x and y axes. Find: (a) the equation of the point's trajectory y(x) , plot this function, (b) the time dependence of the velocity v and acceleration w vectors, as well as of the moduli of these quantities, (c) the time dependence of the angle alpha between the vectors w and v, (d) the mean velocity vector averaged over the first t seconds of motion, and the modulus of this vector.