What is the speed of the particle and what is the magnitude

Question

Q1) At the instant when the particle is at point P on its path (as roughly shown in the XY plane), it possesses the following kinematics data (all in SI units): x' = 1, y' = 2, x'' = -1.5, y'' = 3.5, x = 8, y = 6

i) Express its position, velocity and acceleration in the standard i, j system (with the origin at O). Clearly draw these vectors (i.e. position, velocity, acceleration, i, j) on the figure.

ii) What is the speed of the particle and what is the magnitude of its total acceleration?

iii) Express its velocity and acceleration in the standard e_n, e_t system. Clearly draw e_n, e_t unit vectors on the figure. Find the radius of curvature of the path at this instant.

iv) Express its position, velocity and acceleration in the standard e_r, e_θ system (with the origin at O). Clearly draw er, e_θ unit vectors on the figure.

v) Find the values of r', θ', r'', θ'' as recorded by a stationary observer at O.

Q2) Danica traveling at a speed of 200 km/hr on a straight road applies brakes to her car at point A and reduces her speed at a uniform rate to 150 km/hr at C in a total distance of 300 meters.

a) Calculate the magnitude and direction of the total acceleration of her car immediately after she passes point B. Assume BC to be a part of a circular path. [So, AB is a straight line while BC is a circular arc.]

b) Compute the time she took to go from A to C.

Q3) A disc from point O (as shown) is launched at time t = 0 with an initial speed u at an angle β (0°< β <90°) with respect to the horizontal. It moves in a surrounding medium which offers a constant downward acceleration, say p (p > 0), and a deceleration in the x-direction which is proportional to the horizontal component of its velocity at any instant. Precisely, given this information, one can write the acceleration of the disc as a^→ = -p_j + ax_i where a_x = -qv_x or x'' = qx' >0 (i.e. q is some positive real number). i, j are the standard Cartesian unit vectors.

Find expressions for instantaneous position vector and velocity vector of the disc as functions of time t. Justify whether or not the path of the disc in the medium would be parabolic.