Two Dimensional KinematicsRelative Velocity ProblemsName: _______________________________

1.A jet airliner moving initially at 300 mi/h due east enters a region where the wind is blowing at 100 mi/h in a direction 30.0 north of east. What is the new velocity of the aircraft relative to the ground? (390 mi/h at 7.37 N of E relative to the ground)

The velocity of the plane relative to the ground is the vector sum of the velocity of the plane relative to the air and the velocity of the air relative to the ground, or .

The components of this velocity are

and

Thus,

and

The plane moves at

2.A boat moves through the water of a river at 10 m/s relative to the water, regardless of the boats direction. If the water in the river is flowing at 1.5 m/s, how long does it take the boat to make a round trip consisting of a 300-m displacement downstream followed by a 300-m displacement upstream?

We use the following notation: velocity of boat relative to the shore velocity of boat relative to the water,and velocity of water relative to the shore.

If we take downstream as the positive direction, then for both parts of the trip. Also, while going downstream and for the upstream part of the trip.

The velocity of the boat relative to the shore is given by

While going downstream, and the time to go 300 m downstream is

When going upstream, and the time required to move 300 m upstream is

The time for the round trip is

3.A river flows due east at 1.50 m/s. A boat crosses the river from the south shore to the north shore by maintaining a constant velocity of 10.0 m/s due north relative to the water. (a) What is the velocity of the boat relative to the shore? (b) If the river is 300 m wide, how far downstream has the boat moved by the time it reaches the north shore?

, directed northward, is the velocity of the boat relative to the water.

, directed eastward, is the velocity of the water relative to shore.

is the velocity of the boat relative to shore, and directed at an angle of , relative to the northward direction as shown.

The northward component of is(1)

The eastward component is(2)

(a)Dividing equation (2) by equation (1) gives

From equation (1),

Therefore,

(b)The time to cross the river is and the downstream drift of the boat during this crossing is

4.A rowboat crosses a river with a velocity of 3.30 mi/h at an angle 62.5 north of west relative to the water. The river is 0.505 mi wide and carries an eastward current of 1.25 mi/h. How far upstream is the boat when it reaches the opposite shore?

velocity of boat relative to the water,

velocity of water relative to the shore

and velocity of boat relative to the shore.

as shown in the diagram.

The northward (that is, cross-stream) component of is

The time required to cross the stream is then

The eastward (that is, downstream) component of is

Since the last result is negative, it is seen that the boat moves upstream as it crosses the river. The distance it moves upstream is

(249 ft)

5.How long does it take an automobile traveling in the left lane of a highway at 60.0 km/h to overtake (become even with) another car that is traveling in the right lane at 40.0 km/h when the cars front bumpers are initially 100 m apart?

Choose the positive direction to be the direction of each cars motion relative to Earth. The velocity of the faster car relative to the slower car is given by , where is the velocity of the faster car relative to Earth and is the velocity of Earth relative to the slower car.

Thus, and the time required for the faster car to move 100 m (0.100 km) closer to the slower car is

6.A science student is riding on a flatcar of a train traveling along a straight horizontal track at a constant speed of 10.0 m/s. The student throws a ball along a path that she judges to make an initial angle of 60.0 with the horizontal and to be in line with the track. The students professor, who is standing on the ground nearby, observes the ball to rise vertically. How high does the ball rise? (15.3 m)(7.23 x 103 m, 1.68 x 103 m)

the velocity of the ball relative to the car

velocity of the car relative to Earth

the velocity of the ball relative to Earth

These velocities are related by the equation as illustrated in the diagram.

Considering the horizontal components, we see that

or

From the vertical components, the initial velocity of the ball relative to Earth is

Using , with when the ball is at maximum height, we find

as the maximum height the ball rises.

7.Two canoeists in identical canoes exert the same effort paddling and hence maintain the same speed relative to the water. One paddles directly upstream (and moves upstream), whereas the other paddles directly downstream. With downstream as the positive direction, an observer on shore determines the velocities of the two canoes to be 1.2 m/s and +2.9 m/s, respectively. (a) What is the speed of the water relative to the shore? (b) What is the speed of each canoe relative to the water? (0.85 m/s, 2.1 m/s)

The velocity of a canoe relative to the shore is given by , where is the velocity of the canoe relative to the water and is the velocity of the water relative to shore.

Applied to the canoe moving upstream, this gives

(1)

and for the canoe going downstream

(2)

(a)Adding equations (1) and (2) gives

, so

(b) Subtracting (1) from (2) yields

, or

8.A sailboat is heading directly north at a speed of 20 knots (1 knot = 0.514 m/s). The wind is blowing towards the east with a speed of 17 knots. Determine the magnitude and direction of the wind velocity as measured on the boat. What is the component of the wind velocity in the direction parallel to the motion of the boat? (See Problem 4.54 for an explanation of how a sailboat can move into the wind.) (26 knots at 50 south of east, 20 knots to the south)

We are given that:

(velocity of boat relative to Earth)

and

(velocity of wind relative to Earth)

The velocity of the wind relative to the boat is

where is the velocity of Earth relative to the boat. The vector diagram above shows this vector addition.

Since the vector triangle is a 90 triangle, we find the magnitude of to be

and the direction is given by

Thus,

From the vector diagram above, the component of this velocity parallel to the motion of the boat (that is, parallel to a north-south line) is seen to be