For \(\vec{v}(t)=5t\hat{\imath}-2t^2\hat{\jmath}\), find \(\vec{a}(t)\).

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

If \(a_x=0\) but \(a_y\neq0\), can the path curve?

Find the magnitude of \(\vec{a}=3\hat{\imath}+4\hat{\jmath}\,\mathrm{m\,s^{-2}}\).

A particle's velocity changes from \(2\hat{\imath}+3\hat{\jmath}\) to \(10\hat{\imath}-1\hat{\jmath}\,\mathrm{m\,s^{-1}}\) in \(4\,\mathrm{s}\). Find average acceleration.

For \(\vec{v}=t^2\hat{\imath}+t^2\hat{\jmath}\), find \(\vec{a}\) and its direction at \(t=1\,\mathrm{s}\).

For \(\vec{r}(t)=4t\hat{\imath}+7\hat{\jmath}\), find \(\vec{a}\).

Can an object have acceleration when its speed is constant?

If \(\vec{a}=0\), what can be said about \(\vec{v}\)?

A particle has \(\vec{v}=4\hat{\imath}+3\hat{\jmath}\) and \(\vec{a}=3\hat{\imath}-4\hat{\jmath}\). Is its speed changing at that instant?

For \(\vec{v}=3\hat{\imath}+4t\hat{\jmath}+5t^2\hat{k}\), find \(\vec{a}\) at \(t=2\,\mathrm{s}\).

For \(\vec{v}=6\hat{\imath}\) and \(\vec{a}=-2\hat{\imath}\), is speed increasing or decreasing?

For \(\vec{r}(t)=t^2\hat{\imath}-4t\hat{\jmath}+9\hat{k}\), find \(\vec{a}\).

For \(\vec{a}=6t\hat{\imath}\) and \(\vec{v}(0)=2\hat{\imath}+3\hat{\jmath}\), find \(\vec{v}(t)\).

For \(\vec{v}=-6\hat{\imath}\) and \(\vec{a}=-2\hat{\imath}\), is speed increasing or decreasing? Justify with \(d|\vec v|/dt\).

For \(\vec{r}(t)=\cos t\,\hat{\imath}+\sin t\,\hat{\jmath}\), find \(\vec{a}(t)\).

A velocity vector rotates while keeping the same magnitude. Is acceleration zero?

A particle has \(\vec v=(2t)\hat{\imath}+(t^2-4)\hat{\jmath}\). Find all \(t\) for which speed is stationary \((d|\vec v|/dt=0)\).

A particle has constant \(\vec a=2\hat{\imath}-3\hat{\jmath}\,\mathrm{m\,s^{-2}}\) and \(\vec v(0)=4\hat{\imath}+\hat{\jmath}\,\mathrm{m\,s^{-1}}\). Find the first time when \(\vec v\) is perpendicular to \(\vec a\).

For \(\vec{v}=2t\hat{\imath}-t\hat{\jmath}+4\hat{k}\), find \(\vec{a}\), and compute tangential acceleration \(a_t\) at \(t=1\).

For \(\vec{r}(t)=R\cos(\omega t)\hat{\imath}+R\sin(\omega t)\hat{\jmath}\), derive \(a_t\), \(a_n\), and prove \(|\vec a|=\omega^2R\) even though \(\vec a\) changes direction.

A particle moves in a plane with \(\vec a=-\lambda\vec v+\beta\,\hat{k}\times\vec v\), where \(\lambda,\beta>0\) constants. Derive \(d|\vec v|/dt\), then state and justify whether speed can increase at any time.