Intro to Motion in 2D: Position & Displacement Practice Problems

Suppose that the particle's position depends on time and it moves according to the equation x = (1/3t³ - t²)m and y = (1/4 t²-t)m. Determine where the particle is at (i) t = 0 s and (ii) t = 6 s. Also, find its speed at the given time intervals.

Any vector in a two-dimensional coordinate system can be reduced into its components (x and y). At t = 8s, what should the particle's acceleration vector be if its velocity is v = (4tî + (5-t²)ĵ) m/s?

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 = v cos (kt²) î + v sin (kt²) ĵ, where v = 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.

In the jungle, the position of the monkey in the yz-plane is given by the coordinates

y(t) = At - 0.4 m

z(t) = Bt² m

where A = 1.1 m/s and B = 0.95 m/s².
Sketch the path that the monkey takes for 10 seconds from its initial position.

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^-10 m) and the time t is in seconds.

A hiker starts at a campsite and walks 15 km north, then turns and walks 11 km west, and finally, walks 8.0 km south. Calculate the hiker's final displacement from the campsite. Assume the distances are measured in a straight line.

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.

At t= 0, a submarine is moving at v = -8.0 ĵ m/s and is located at r₀ = 5000 î + 125 ĵ + 750 k̂  ( in meters) with respect to a reference island. At t= 0 s, the submarine accelerates at a rate of a = 1.25  î - 2.50  ĵ (m/s²) for 2 minutes. Calculate the submarine's position vector after 2 minutes of motion.

Determine the resultant vector by graphically adding the following vector displacements:

(i) 30 m, 45° south of west; (ii) 22 m, 60° west of north; (iii) 15 m, 30° east of north; (iv) 38 m, 75° south of east.

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.

From a base camp, a mountain climber begins her ascent, moving 800 m towards the north and then 600 m towards the east to reach an intermediate camp. The next day, she climbs 400 m vertically to reach the summit. Expressing your answer in components, determine the displacement of the climber from the base camp. Additionally, in both the vertical and horizontal planes, get the magnitude and angles with respect to the 𝓍-axis. Consider 𝓍 is east, y is north, and 𝒵 is up.