Direction of Instantaneous Velocity

Q.

The position $\overrightarrow{r}=(2.00t^3-5.00t)\hat{i}+(6.00-7.00t^4)\hat{j},with\overrightarrow{r}$ in meters and t in seconds.
What is the angle between the positive direction of the x-axis and a line tangent to the particle's path at t=2.00 s?

1. $ta{n}^{-1}\left(\frac{28}{6}\right)$

2. $ta{n}^{-1}\left(\frac{-224}{19}\right)$

3. $ta{n}^{-1}\left(\frac{1}{\sqrt{3}}\right)$

4. ${45}^{\circ }$

Q.

The position $\overrightarrow{r}=(2.00t^3-5.00t)\hat{i}+(6.00-7.00t^4)\hat{j},with\overrightarrow{r}$ in meters and t in seconds.
What is the angle between the positive direction of the x-axis and a line tangent to the particle's path at t=2.00 s?

1. $ta{n}^{-1}\left(\frac{28}{6}\right)$

2. $ta{n}^{-1}\left(\frac{-224}{19}\right)$

3. $ta{n}^{-1}\left(\frac{1}{\sqrt{3}}\right)$

4. ${45}^{\circ }$

Q.

The position vector $\overrightarrow{r}=5.00t\hat{i}+(et+f{t}^2)\hat{j}$ locates a particle as a function of time t. Vector $\overrightarrow{r}$ is in meters, t is in seconds, and factors e and f are constants. Below Diagram gives the angle $\theta$ of the particle's direction of travel as a function of t($\theta$) is measured from the positive x direction). What are 'e' and 'f', including units?

1. $e=\frac{35}{6\sqrt{3}}m/s,f=\frac{-5}{24\sqrt{3}}m/s^2$

2. $e=\frac{-35}{6\sqrt{3}}m/s,f=\frac{5}{24\sqrt{3}}m/s$

3. $e=\frac{35}{6\sqrt{3}}m/s,f=\frac{5}{24\sqrt{3}}m/s^2$

4. $e=\frac{35}{6\sqrt{3}}m/s,f=\frac{5}{24\sqrt{3}}m/s^2$