4. 2D Kinematics

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

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.

(II) An airplane is traveling 815 km/h in a direction 41.5° west of north (Fig. 3–40). How far north and how far west has the plane traveled after 1.75 h? 

A delivery truck travels 21 blocks north, 16 blocks east, and 26 blocks south. What is its final displacement from the origin? Assume the blocks are equal length.

The coordinates of a bird flying in the xy-plane are given by x(t) = αt and y(t) = 3.0 m − βt², where α = 2.4 m/s and β = 1.2 m/s². (a) Sketch the path of the bird between t = 0 and t = 2.0 s.

The position of a squirrel running in a park is given by $\vec{r} = \left[ (0.280~\mathrm{m/s})t + (0.0360~\mathrm{m/s^2})t^2 \right] \hat{i} + (0.0190~\mathrm{m/s^3})t^3 \hat{j}$. At $t=5.00s$, how far is the squirrel from its initial position?

A particle's trajectory is described by $x=(\frac{1}{2}{t}^3-2t^2)\text{m}\text{and}y=(\frac{1}{2}{t}^2-2t)\text{m},$ where $t$ is in $s$. What are the particle's position and speed at $t=0\text{ s}$ and $t=4\text{ s}$?

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?

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 $\mathrm{km}$.

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?

CALC A 100 g bead slides along a frictionless wire with the parabolic shape y = (2m^-1) x². Find an expression for a_y, the vertical component of acceleration, in terms of x, v_x, and a_x. Hint: Use the basic definitions of velocity and acceleration.

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 web page designer creates an animation in which a dot on a computer screen has position $\overrightarrow{r}=[4.0\mathrm{cm}+\left(2.5{\text{ cm/s}}^2\right)t^2]i+\left(5.0\mathrm{cm/s}\right)tj$. Find the magnitude and direction of the dot's average velocity between $t=0$ and $t=2.0s$.