A particle moves from \((1,2)\,\mathrm{m}\) to \((5,-1)\,\mathrm{m}\). Find \(\Delta\vec{r}\).

Write the position vector for a particle at \(x=-3\,\mathrm{m}\), \(y=4\,\mathrm{m}\), \(z=2\,\mathrm{m}\).

If \(z(t)=0\) for all \(t\), what can be said about the motion?

For \(\vec{r}(t)=2t\hat{\imath}+3t^2\hat{\jmath}\), find \(\vec{v}(t)\).

For \(\vec{v}=6\hat{\imath}+8\hat{\jmath}\,\mathrm{m\,s^{-1}}\), find the speed.

A particle has \(\vec{v}=v_x\hat{\imath}+v_y\hat{\jmath}\). Write its speed.

A particle's displacement is \(12\hat{\imath}-5\hat{\jmath}\,\mathrm{m}\) over \(4\,\mathrm{s}\). Find average velocity.

For \(\vec{r}(t)=t^2\hat{\imath}+(5-t)\hat{\jmath}\), find \(\vec{v}\) at \(t=2\,\mathrm{s}\).

For \(\vec{r}(t)=(t^2-4t)\hat{\imath}+3\hat{\jmath}\), when is the velocity zero?

A particle travels around a semicircle of radius \(2\,\mathrm{m}\). Its displacement magnitude is \(4\,\mathrm{m}\). What distance did it travel?

A particle has \(\vec{v}=3\hat{\imath}-4\hat{\jmath}+12\hat{k}\,\mathrm{m\,s^{-1}}\). Find its speed and direction cosine with the \(+\hat{k}\)-axis.

If \(\vec{r}(t)=4\cos t\,\hat{\imath}+4\sin t\,\hat{\jmath}\), find \(\vec{v}(t)\) and show that \(\vec{r}\cdot\vec{v}=0\).

Can two particles have the same speed but different velocities?

Show why shifting the origin by a constant vector \(\vec{c}\) does not change displacement.

A particle moves as \(\vec{r}(t)=(t^2-4t)\hat{\imath}+(t+1)\hat{\jmath}\). Find the instant of closest approach to the origin and the minimum distance.

For \(\vec{r}(t)=(t^2-1)\hat{\imath}+2t\hat{\jmath}+3\hat{k}\), find \(\vec{v}(t)\) and the angle between \(\vec{v}(1)\) and \(\hat{\imath}\).

A particle's velocity is tangent to its path. What does that mean physically?

A particle starts at \(\vec{r}_0=2\hat{\imath}-\hat{\jmath}\,\mathrm{m}\) and has velocity \(\vec{v}(t)=(4t-1)\hat{\imath}+(3-2t)\hat{\jmath}\). Find \(\vec{r}(2)\).

A trajectory is \(\vec{r}(t)=t\hat{\imath}+(t^2-2t)\hat{\jmath}\). Find the equation of the tangent line at \(t=3\).

For \(\vec{r}(t)=e^t\hat{\imath}+t^3\hat{\jmath}\), find \(\vec{v}(t)\) and determine whether speed is increasing at \(t=0\).

A particle has \(\vec{r}(t)=(t^2-2t)\hat{\imath}+(3t-t^2)\hat{\jmath}\). Find all times when the velocity is perpendicular to position, and classify each event as moving toward or away from the origin immediately after that instant.

A particle moves with \(\vec{r}(t)=R\cos(\omega t)\hat{\imath}+R\sin(\omega t)\hat{\jmath}+bt\hat{k}\). Prove speed is constant, derive curvature \(\kappa\), and state how \(\kappa\) changes as \(b\) increases with \(R,\omega\) fixed.