4. 2D Kinematics

An object starts at position $x=0m$ and travels with a constant velocity of $v=3\frac{m}{s}$ along the x-axis. What is its position after $4s$?

A particle's trajectory is described by $x = \(\left\)(\(\frac{1}{2}\) t^3 - 2t^2\(\right\)) \, \(\text{m}\) \(\quad\) \(\text{and}\) \(\quad\) y = \(\left\)(\(\frac{1}{2}\) t^2 - 2t\(\right\)) \, \(\text{m}\),$ where $t$ is in $s$. What is the particle's direction of motion, measured as an angle from the $x$-axis, at $t=0\(\text{ s}\)$ and $t=4\(\text{ s}\)$?

Graphically determine the resultant of the following three vector displacements: 26 m, 33° west of south.

Which of the following is a simple model commonly used to help understand the forces acting on an object in two-dimensional motion?

Ignoring friction and air resistance, if you start from rest and accelerate at a constant rate of $2m/s^2$, how fast will you be moving after $10s$?

You have a remote-controlled car that has been programmed to have velocity $v=(-3ti+2t^{2}j)\text{m/s}$, where t is in s. At t = 0 s, the car is at $r_0=(3.0i+2.0j)\text{m}$. What are the car's position vector?

At point A, a hiker is 10m east from the origin. After 35s, the hiker arrives at point B 40m at 60° north of east from the origin. Calculate the magnitude and direction of the hiker's displacement.

In a two-dimensional pulley system, a block moves horizontally to the right by $3m$ and then vertically upward by $4m$. What is the magnitude of the block's total displacement?

A plumber steps out of his truck, walks 55 m east and 38 m south, and then takes an elevator 12 m into the subbasement of a building where a bad leak is occurring. What is the displacement of the plumber relative to his truck? Give your answer in components. Also give the magnitude and angles, with respect to the 𝓍 axis, in the vertical and horizontal planes. Assume 𝓍 is east, y is north, and 𝒵 is up.

A wheel starts from rest and rotates with a constant angular acceleration until it comes to a stop after reaching an angular velocity of $\omega =0$ at time $t$. If its initial angular velocity was $\omega ={\omega }_0$ at $t=0$, through what total angle $\theta$ did the wheel turn between $t=0$ and the time it stopped?

A spaceship maneuvering near Planet Zeta is located at $r=(600i-400j+200k)\times {10}^3\text{km},$ relative to the planet, and traveling at $v=9500i\text{m/s}$. It turns on its thruster engine and accelerates with $a=(40i-20k){\text{m/s}}^2$ for $35\text{ min}$. What is the spaceship's position when the engine shuts off? Give your answer as a position vector measured in $\mathrm{km}$.

A particle moving in the xy-plane has velocity v = (2ti + (3-t²)j) m/s, where t is in s. What is the particle's acceleration vector at t = 4s?

Scientists design a new particle accelerator in which protons (mass 1.7 X 10^-27 kg) follow a circular trajectory given by $r=c\cos \left(k{t}^2\right)i+c\sin \left(k{t}^2\right)j$ where c = 5.0 m and k = 8.0 x 10⁴ rad/s² are constants and t is the elapsed time. What is the radius of the circle?

A $2.00$-kg box is moving to the right with speed $9.00$ m/s on a horizontal, frictionless surface. At $t=0$ a horizontal force is applied to the box. The force is directed to the left and has magnitude $F\left(t\right)=$ ($6.00$ N/s²)t². What distance does the box move from its position at $t=0$ before its speed is reduced to zero?