91Ó°ÊÓ

A small plane is flying directly west with an airspeed of $30.0 \mathrm{m} / \mathrm{s} .\( The plane flies into a region where the wind is blowing at \)10.0 \mathrm{m} / \mathrm{s}\( at an angle of \)30^{\circ}$ to the south of west. In that region, the pilot changes the directional heading to maintain her due west heading. (a) What is the change she makes in the directional heading to compensate for the wind? (b) After the heading change, what is the ground speed of the airplane?

A boat that can travel at \(4.0 \mathrm{km} / \mathrm{h}\) in still water crosses a river with a current of \(1.8 \mathrm{km} / \mathrm{h}\). At what angle must the boat be pointed upstream to travel straight across the river? In other words, in what direction is the velocity of the boat relative to the water?

At an antique car rally, a Stanley Steamer automobile travels north at $40 \mathrm{km} / \mathrm{h}\( and a Pierce Arrow automobile travels east at \)50 \mathrm{km} / \mathrm{h}$. Relative to an observer riding in the Stanley Steamer, what are the \(x\) - and \(y\) -components of the velocity of the Pierce Arrow car? The \(x\) -axis is to the east and the \(y\) -axis is to the north.

Sheena can row a boat at \(3.00 \mathrm{mi} / \mathrm{h}\) in still water. She needs to cross a river that is 1.20 mi wide with a current flowing at $1.60 \mathrm{mi} / \mathrm{h} .$ Not having her calculator ready, she guesses that to go straight across, she should head \(60.0^{\circ}\) upstream. (a) What is her speed with respect to the starting point on the bank? (b) How long does it take her to cross the river? (c) How far upstream or downstream from her starting point will she reach the opposite bank? (d) In order to go straight across, what angle upstream should she have headed?

A dolphin wants to swim directly back to its home bay, which is $0.80 \mathrm{km}\( due west. It can swim at a speed of \)4.00 \mathrm{m} / \mathrm{s}$ relative to the water, but a uniform water current flows with speed \(2.83 \mathrm{m} / \mathrm{s}\) in the southeast direction. (a) What direction should the dolphin head? (b) How long does it take the dolphin to swim the \(0.80-\mathrm{km}\) distance home?

Demonstrate with a vector diagram that a displacement is the same when measured in two different reference frames that are at rest with respect to each other.

A boy is attempting to swim directly across a river; he is able to swim at a speed of \(0.500 \mathrm{m} / \mathrm{s}\) relative to the water. The river is \(25.0 \mathrm{m}\) wide and the boy ends up at \(50.0 \mathrm{m}\) downstream from his starting point. (a) How fast is the current flowing in the river? (b) What is the speed of the boy relative to a friend standing on the riverbank?

Jason is practicing his tennis stroke by hitting balls against a wall. The ball leaves his racquet at a height of \(60 \mathrm{cm}\) above the ground at an angle of \(80^{\circ}\) with respect to the vertical. (a) The speed of the ball as it leaves the racquet is \(20 \mathrm{m} / \mathrm{s}\) and it must travel a distance of \(10 \mathrm{m}\) before it reaches the wall. How far above the ground does the ball strike the wall? (b) Is the ball on its way up or down when it hits the wall?

Imagine a trip where you drive along an east-west highway at $80.0 \mathrm{km} / \mathrm{h}$ for 45.0 min and then you turn onto a highway that runs \(38.0^{\circ}\) north of east and travel at \(60.0 \mathrm{km} / \mathrm{h}\) for 30.0 min. (a) What is your average velocity for the trip? (b) What is your average velocity on the return trip when you head the opposite way and drive \(38.0^{\circ}\) south of west at \(60.0 \mathrm{km} / \mathrm{h}\) for the first \(30.0 \mathrm{min}\) and then west at \(80.0 \mathrm{km} / \mathrm{h}\) for the last \(45.0 \mathrm{min} ?\)