A point moves in the plane so that its tangential acceleration `w_tau=a`, and its normal acceleration `w_n=bt^4`, where a and b are positive constants, and t is time. At the moment `t=0` the point was at rest. Find how the curvature radius R of the point's trajectory and the total acceleration w depend on the distance covered s.

A point moves in the plane xy according to the law x=a sin omegat , y=a(1-cos omega t) , where a and omega are positive constants. Find: (a) the distance s traversed by the point during the time tau ,

A point moves in the plane xy according to the law x = alpha sin omega t, y = alpha(1 - cos omega t) , where alpha and omega are positive constant and t is time. Find the distance traversed by point in time t_(0) .

A point moves in xy-plane according to equation x = at, y = at (l - bt) where a and b are positive constants and t is time. The instant at which velocity vector is at pi//4 with acceleration vector is given by:

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.

A particle moves in a circular path of radius R with an angualr velocity omega=a-bt , where a and b are positive constants and t is time. The magnitude of the acceleration of the particle after time (2a)/(b) is

A particle moves in a circular path of radius R with an angualr velocity omega=a-bt , where a and b are positive constants and t is time. The magnitude of the acceleration of the particle after time (2a)/(b) is

A point moves in the plane xy according to the law x=at , y=at(1-alphat) , where a and alpha are positive constants, and t is time. Find: (a) the equation of the point's trajectory y(x) , plot this function, (b) the velocity v and the acceleration w of the point as functions of time, (c) the moment t_0 at which the velocity vector forms an angle pi//4 with the acceleration vector.

A point moves in the plane xy according to the law x=a sin omegat , y=a(1-cos omega t) , where a and omega are positive constants. Find: (a) the distance s traversed by the point during the time tau , (b) the angle between the point's velocity and acceleration vectors.

A point moves with decleration along the circle of radius R so that at any moment of time its tangential and normal accelerations are equal in moduli. At the initial moment t=0 the velocity of the point equals v_0 . Find: (a) the velocity of the point as a function of time and as a function of the distance covered s_1 , (b) the total acceleration of the point as a function of velocity and the distance covered.