A boat moves east at \(4\,\mathrm{m\,s^{-1}}\) relative to water. The river flows north at \(3\,\mathrm{m\,s^{-1}}\). Find boat velocity relative to ground.

Using \(\vec{v}_{BG}=4\hat{\imath}+3\hat{\jmath}\,\mathrm{m\,s^{-1}}\), find ground speed.

If \(\vec{v}_{AB}=7\hat{\imath}\,\mathrm{m\,s^{-1}}\), what is \(\vec{v}_{BA}\)?

A person walks forward at \(1.2\,\mathrm{m\,s^{-1}}\) on a train moving forward at \(15\,\mathrm{m\,s^{-1}}\). Find person velocity relative to ground.

A person walks backward at \(1.2\,\mathrm{m\,s^{-1}}\) on a train moving forward at \(15\,\mathrm{m\,s^{-1}}\). Find person velocity relative to ground.

A boat aims due north at \(6\,\mathrm{m\,s^{-1}}\) relative to water. The current is east at \(2\,\mathrm{m\,s^{-1}}\). Find drift angle east of north.

A plane has airspeed \(80\,\mathrm{m\,s^{-1}}\) east. Wind is \(20\,\mathrm{m\,s^{-1}}\) west. Find ground velocity.

A plane has airspeed \(80\,\mathrm{m\,s^{-1}}\) east. Wind is \(20\,\mathrm{m\,s^{-1}}\) north. Find ground speed.

A drone moves \(5\,\mathrm{m\,s^{-1}}\) north relative to air. Wind is \(2\,\mathrm{m\,s^{-1}}\) south. Find ground velocity.

A plane wants a ground velocity due east at \(100\,\mathrm{m\,s^{-1}}\). Wind is north at \(20\,\mathrm{m\,s^{-1}}\). What must the plane's air-velocity components be?

If \(\vec{v}_{PA}=\vec{v}_{PB}+\vec{v}_{BA}\), solve for \(\vec{v}_{PB}\).

A swimmer can swim \(2\,\mathrm{m\,s^{-1}}\) north relative to water. Current is \(1\,\mathrm{m\,s^{-1}}\) east. Find swimmer velocity relative to ground.

A walkway moves east at \(1\,\mathrm{m\,s^{-1}}\). A person walks west at \(1.5\,\mathrm{m\,s^{-1}}\) relative to the walkway. Find ground velocity, east positive.

A rain drop falls vertically at \(8\,\mathrm{m\,s^{-1}}\) relative to air. Wind blows east at \(6\,\mathrm{m\,s^{-1}}\). Find rain velocity relative to ground.

A boat has speed \(4\,\mathrm{m\,s^{-1}}\) relative to water. Current is \(3\,\mathrm{m\,s^{-1}}\) east. Find all possible ground-speed magnitudes and the headings that produce minimum and maximum values.

Car A moves east at \(25\,\mathrm{m\,s^{-1}}\). Car B moves north at \(15\,\mathrm{m\,s^{-1}}\). Find \(\vec v_{AB}\), its magnitude, and its direction measured from east toward north.

Why must relative velocities be added as vectors?

A pilot wants a ground track due north at \(120\,\mathrm{m\,s^{-1}}\). Wind is \(35\,\mathrm{m\,s^{-1}}\) east. Find required heading angle and required airspeed.

If frame B moves with constant velocity relative to A, derive the relation between \(\vec a_{PA}\) and \(\vec a_{PB}\). Then state what changes if B has constant nonzero acceleration relative to A.

For \(\vec r_{PA}=6\hat\imath+2\hat\jmath\,\mathrm m\), \(\vec r_{BA}=4\hat\imath-\hat\jmath\,\mathrm m\), and \(\vec v_{BA}=(-2\hat\imath+3\hat\jmath)\,\mathrm{m\,s^{-1}}\), find \(\vec r_{PB}\) and \(\vec v_{PB}\) given \(\vec v_{PA}=5\hat\imath+\hat\jmath\,\mathrm{m\,s^{-1}}\).

A swimmer speed relative to water is \(2.5\,\mathrm{m\,s^{-1}}\). River current is \(1.0\,\mathrm{m\,s^{-1}}\) east, river width \(120\,\mathrm m\). Compare (i) minimum-time crossing and (ii) zero-drift crossing: find heading, crossing time, and downstream drift in each strategy.

An aircraft has fixed airspeed \(200\,\mathrm{m\,s^{-1}}\). Wind is \(60\,\mathrm{m\,s^{-1}}\) from west to east. The aircraft flies out-and-back along an east-west line over equal ground distances. Derive total travel time and average ground speed for the round trip, and compare to no-wind case.