4. 2D Kinematics

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 $t=2.0s$.

The position of a squirrel running in a park is given by $\vec{r} = \left[ (0.280~\mathrm{m/s})t + (0.0360~\mathrm{m/s^2})t^2 \right] \hat{i} + (0.0190~\mathrm{m/s^3})t^3 \hat{j}$. (a) What are $v_x\left(t\right)$ and $v_y\left(t\right)$, the $x$-and $y$-components of the velocity of the squirrel, as functions of time?

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. Calculate the velocity and acceleration vectors of the bird as functions of time.

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 $t=4\text{ s}$?

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?