For \(x(t)=2t^3-t\), find \(v_x\), \(a_x\), and their values at \(t=1\,\mathrm{s}\).

If \(v_x(t)=4t\) and \(x(0)=3\,\mathrm{m}\), find \(x(t)\).

For \(x(t)=t^3-6t^2\), find when acceleration is zero.

If \(a_x(t)=6t\) and \(v_x(0)=2\,\mathrm{m\,s^{-1}}\), find \(v_x(t)\).

For \(v_x(t)=3t^2-2\), find displacement from \(t=0\) to \(t=2\,\mathrm{s}\).

If \(v_x(t)=kt\) and \(x(0)=0\), find \(x(t)\).

For \(a_x(t)=2t+1\), find the velocity change from \(t=0\) to \(t=3\,\mathrm{s}\).

For \(x(t)=t^4\), find \(v_x(t)\) and \(a_x(t)\).

For \(x(t)=5+3t-2t^2\), find when the particle stops.

What physical quantity is represented by area under a velocity-time graph?

For \(x(t)=A\sin(\omega t)\), find \(v_x(t)\) and \(a_x(t)\).

If \(v_x(t)=5\,\mathrm{m\,s^{-1}}\) and \(x(0)=x_0\), find \(x(t)\).

For \(v_x(t)=6-2t\), find displacement from \(t=0\) to \(t=5\,\mathrm{s}\).

Check the units of \(\int v_x(t)\,dt\).

If velocity is constant, what is acceleration?

Why are initial conditions needed after integration?

If \(a_x=4\,\mathrm{m\,s^{-2}}\) and \(v_x(2)=10\,\mathrm{m\,s^{-1}}\), find \(v_x(t)\).

For \(v_x(t)=t^2\), find displacement from \(t=1\,\mathrm{s}\) to \(t=3\,\mathrm{s}\).

Starting with \(a_y=-g\), integrate to get the free-fall velocity and position equations.

What physical quantity is represented by area under an acceleration-time graph?

A particle has \(a_x(t)=6t-4\), \(v_x(0)=3\,\mathrm{m\,s^{-1}}\), and \(x(0)=-2\,\mathrm{m}\). Find \(v_x(t)\) and \(x(t)\).

A particle has \(v_x(t)=t^2-4t+3\). Find its displacement and distance traveled from \(t=0\) to \(t=4\,\mathrm{s}\).