Motion in Two or Three Dimensions

Problems for solution without assistance

Problem 1

A military airplane on a routine training mission is flying horizontally at a speed of 120 m/s and accidently drops a bomb (fortunately not armed) at an elevation of 2000 m. Air resistance may be ignored, a) How much time is required for the bomb to reach the earth? b) How far does it travel horizontally while falling? c) Find the horizontal and vertical components of its velocity just before it strikes the earth, d) Where is the airplane when the bomb strikes the earth if the velocity of the airplane remains constant?

Problem 2

A sharpshooter fires a 0.22-caliber rifle horizontally at a target. The bullet has a muzzle velocity of magnitude 275 m/s. Air resistance may be ignored, a) How far does the bullet drop in flight if the target is 75 m away? b) Sketch a graph of the vertical drop of the bullet as a function of the distance to the target.

Problem 3

Warren Moon throws a football with an initial upward velocity component of 15.0 m/s and a horizontal velocity component of 25.0 m/s. Air resistance may be ignored, a) How much time is required for the football to reach the highest point of the trajectory?
b) How high is this point?
c) How much time (after being thrown) is required for the football to return to its original level? How does this compare with the time calculated in part a)?
d) How far has it traveled horizontally during this time?

Problem 4

During a game in 1982, Reggie Jackson threw a baseball at an angle of 53.1° above the horizontal with an initial speed 40.0 m/s. Air resistance may be ignored,
a) At what two times was the baseball at a height of 25.0 m above the point from which it was thrown?
b) Calculate the horizontal and vertical components of the baseball's velocity at each of the two times calculated in part (a),
c) What were the magnitude and direction of the baseball's velocity when it returned to the level from which it was thrown?

Problem 5

A pistol that fires a signal flare gives the flare an initial speed (muzzle speed) of 180 m/s. Air resistance may be ignored, a) If the flare is fired at an angle of 55° above the horizontal on the level salt flats of Utah, what is its horizontal range? b) If the flare is fired at the same angle over the flat Sea of Tranquility on the moon, where g = 1.6 m/s², what is its horizontal range?

Problem 6

A Civil War mortar called the Dictator fired its 90.7-kg (200-lb) shell a maximum horizontal distance of 4345 m (4752 yd) when tbe shell was projected at an angle of 45° above the horizontal. Air resistance may be ignored,
a) What was the muzzle speed of the shell (the speed of the shell as it left the barrel of the mortar)? b) What maximum height above the ground did the shell reach?
c) For what amount of time was the shell in the air?

Problem 7

A man stands on the roof of a building that is 30.0 m tall and throws a rock with a velocity of magnitude 40.0 m/s at an angle of 33.0° above the horizontal. Air resistance may be ignored. Calculate a) the maximum height above the roof reached by the rock; b) the magnitude of the velocity of the rock just before it strikes the ground; c) the horizontal distance from the base of the building to the point where the rock strikes the ground.

Problem 8

A man stands on the roof of a building that is 30.0 m tall and throws a rock with a velocity of magnitude 40.0 m/s at an angle of 33.0° above the horizontal. Air resistance may be ignored. Calculate a) the maximum height above the roof reached by the rock; b) the magnitude of the velocity of the rock just before it strikes the ground; c) the horizontal distance from the base of the building to the point where the rock strikes the ground.

Problem 9

On your first day at work for an appliance manufacturer, you are told to figure out what to do to the period of rotation during a washer's spin cycle to double the centripetal acceleration. You impress your boss by answering immediately. What do you tell her?

Problem 10

The earth has a radius of 6.38 x 10⁶ m and turns around once on its axis in 24 h.
a) What is the radial acceleration of an object at the earth's equator? Give your answer in m/s² and as a fraction of g. b) If a_rad at the equator is greater than or equal to g, objects would fly off the earth's surface and into space. What would the period of the earth's rotation have to be for this to occur?

Problem 11

The radius of the earth's orbit around the sun (assumed to be circular) is 1.50 x 10¹¹ m, and the earth travels around this orbit in 365 days,
a) What is the magnitude of the orbital velocity of the earth in m/s?
b) What is the radial acceleration of the earth toward the sun in m/s²?

Problem 12

A Ferris wheel with radius 14.0 m is turning about a horizontal axis through its center. The linear speed of a passenger on the rim is constant and equal to 8.00 m/s.
a) What are the magnitude and direction of the passenger's acceleration as she passes through the lowest point in her circular motion?
b) How much time does it take the Ferris wheel to make one revolution?

Problem 13

A railroad flatcar is traveling to the right at a speed of 13.0 m/s relative to an observer standing on the ground. A motor scooter is being ridden on the flatcar. What is the velocity (magnitude and direction) of the motor scooter relative to the flatcar if its velocity relative to the observer on the ground is
a) 20.0 m/s to the right?
b) 4.0 m/s to the left?
c) zero?

Problem 14

A "moving sidewalk" in an airport terminal building moves at 1.0 m/s and is 40.0 m long. If a woman steps on at one end and walks at 2.0 m/s relative to the moving sidewalk, how much time does she require to reach the opposite end if she walks
a) in the same direction the sidewalk is moving?
b) in the opposite direction?

Problem 15

A canoe has a velocity of 0.30 m/s northwestrelative to the earth. The canoe is on a river that is flowing 0.50 m/s west relative to the earth. Find the velocity (magnitude and direction) of the canoe relative to the river.

Problem 16

An airplane pilot wishes to fly due north. A wind of 80.0 km/h (about 50 mi/h) is blowing toward the west,
a) If the airspeed of the plane (its speed in still air) is 240.0 km/h (about 150 mi/h), in what direction should the pilot head?
b) What is the speed of the plane over the ground? Illustrate with a vector diagram.

Problem 17

A river flows due north with a speed of 2.4 m/s. A man rows a boat across the river; his velocity relative to the water is 4.2 m/s due east. The river is 1000 m wide,
a) What is his velocity relative to the earth?
b) How much time is required to cross the river?
c) How far north of his starting point will he reach the opposite bank?

Examples of solutions

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