Problems

practice

1. A laser beam is aimed 15.95° above the horizontal at a mirror 11,648 m away. It glances off the mirror and continues for an additional 8570. m at 11.44° above the horizon until it hits its target. What is the resultant displacement of the beam to the target?
2. Three forces act on a point: 3 N at 0°, 4 N at 90°, and 5 N at 217°. What is the net force?
3. A cyclist heads due west on a straight road. The wind is blowing from 248° at 10 m/s.
   1. Is this wind more like a headwind or a tailwind?
   2. What is the headwind/tailwind speed?
   3. What is the crosswind speed?
4. One unfortunate winter day I happened to slip on an icy ramp inclined 37° to the horizontal. Find my acceleration down the ramp given that the acceleration due to gravity points straight down and has a value of 9.8 m/s². (Assume the ice is perfectly frictionless.)

conceptual

1. Prove that the angle between the surface of an inclined plane and a the horizontal (the angle of inclination) is equal to the angle between the normal of the plane and the vertical.

numerical

1. A utility pole is held upright with a single guy wire under 3500 N of tension. The guy wire meets the ground at a 60° angle. What is the magnitude of the component of this force that …
   1. prevents the pole from tipping over?
   2. keeps the pole in contact with the earth?
2. A force of 90 N is applied to a lawnmower along a handle that makes a 30° angle with the vertical. What is the magnitude of the component of this force that …
   1. propels the lawnmower?
   2. keeps the lawnmower in contact with the ground?
3. A boy pulls a sled along level ground by means of a rope under 78 N of tension. The rope with which he pulls the sled makes an angle of 37° with the horizontal. What is the magnitude of the component of the tension that …
   1. actiually moves the sled?
   2. is wasted?
4. A cyclist heads out on a straight road with a bearing of 60° east of north. A north wind is blowing at 10 m/s.
   1. Is this wind more like a headwind or a tailwind?
   2. What is the effective speed of this headwind/tailwind?
5. A cyclist heads out on a straight road with a bearing of 60° east of north. A southwest wind is blowing at 10 m/s.
   1. Is this wind more like a headwind or a tailwind?
   2. What is the effective speed of this headwind/tailwind?
6. Three of the four busiest airports in the US are Hartsfield-Jackson in Atlanta, Georgia (ATL), O'Hare in Chicago, Illinois (ORD), and Dallas/Fort Worth in Texas (DFW). Complete the following table for roundtrip airplane journey from Atlanta to Chicago to Dallas to Atlanta. Note: Headings are measured clockwise from due north.

   departure	arrival	distance	heading
   ATL	ORD	975 km	342.8°
   ORD	DFW	1291 km	221.6°
   DFW	ATL
   round trip			n/a

worksheets

1. worksheet-components.pdf
   Calculate the magnitude (in centimeters) and direction (in standard form) of the resultant vector for each diagram using the method of components. Use the 1 cm gridlines towards this end. A ruler and protractor are not needed for this exercise.
2. worksheet-words.pdf
   A copy of all the word problems in this section are also available on a pdf worksheet.