A private tech company built one of the most advanced and powerful particle accelerators ever. The accelerator uses electricity to 'push' the charged particles (electrons) along a circular path, making them go faster and faster. The path they follow is given by the equation r = R$\(\cdot\)$cos(kt²) î + R.sin(kt²) ĵ, where R = 4 m and k = 7 x 10⁴ rad/s² are constants and t is the time. Determine the circle's radius.

Consider a projectile's (m = 150 g) parabolic trajectory given by the equation y = (4m⁻¹) x². Using the standard definitions of velocity and acceleration, find an expression that expresses a_y, the acceleration's vertical component, in terms of x, v_x, and a_x.

The position of a hypothetical quark moving in sub-atomic space is given by

Calculate how far the hypothetical quark is from its initial position after a period of 4 seconds. The unit of the function s is in angstroms (10^-10m) and the time t is in seconds.

A car is traveling on an interstate highway. The x- and y-components of the car's velocity are given by the equation v = (2tî  + 9t² ĵ) m/s. Assuming that the car's initial position (r₀) at a time t_o is given as r₀ = (5î + 3ĵ)m, determine its position vector at t = 3s.

A car is traveling at a speed of 135 km/h in a direction 29.7° north of east. Calculate the distances covered by the car in the northward and eastward directions after 3.3 hours.