On the axes shown in Figure 2, sketch a graph of the y-component of the puck’s velocity $v_y$ as a function of time $t$ from $t = 0$ to $t = 4.0\ \text{s}$, as measured in the lab frame.

Derive an expression for the x- and y-components of the puck’s displacement $\Delta \vec{r}$ during the interval from $t = 0$ to $t = 4.0\ \text{s}$ as measured in the lab frame. Express your answers in terms of the given speed 2.0 m/s, the angle $30^\circ$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Derive an expression for the magnitude of the puck’s average velocity $\left|\vec{v}_{\text{avg}}\right|$ over the interval from $t = 0$ to $t = 4.0\ \text{s}$ as measured in the lab frame. Express your answer in terms of the given speed 2.0 m/s and the angle $30^\circ$. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.