Unit 3

163 Your Turn Solutions


1. Explain what is meant by the elephant rule for vector addition.

When adding vectors, it is necessary to place the head (trunk) of one vector on the tail of another. The resultant is drawn from the tail of the first vector to the trunk (head) of the last.

2. When a vector is to be added graphically to a second vector, which of the following may you do to the first vector: move it, rotate it, or change its length?

When adding vectors, it is often necessary to move one or more of them. It is not appropriate to rotate or increase the length of any vector since this actually changes the vector.

3. Define the term resultant.

A resultant vector is the sum, or the vector that results from vector addition.

4. Two vectors with magnitudes of 50 meters and 70 meters respectively.

a. The maximum displacement that can be achieved with these two vectors is 120m.

b. The minimum displacement that can be achieved with these two vectors is 20m.

 

5. Mike Hiker walks 11 km due north from camp, then 11 km due east.

a. What is Mike's total distance walked?

The total distance is the sum of the distances without concern as to direction. 22 km.

b. What is Mike's total displacement from camp?

The total displacement is 16 km @ 45o

6. Kara Lot has to find her lost younger brother. He yells out that he walked 70 meters due south, 60 meters at 225 degrees and finally walked 50 meters at 180 degrees. What is the minimum distance Kara can walk to locate her brother and in what direction should she walk?

Kara should walk 146 km @ 231o.

7. A hot air balloon flies in a 20 mile per hour wind. What speed does the balloon measure as its speed with respect to:

a. the wind?

The balloon will measure its speed with respect to the air to be 0 m/s.

b. the ground?

The balloon will measure its speed with respect to the ground to be 20 m/s.

 

8. A boat points straight East across a river and measures its velocity across the river to be 15 m/s, East. If it is in a river that travels at 10 m/s, South, what is the boat's:

a. Eastward velocity?

The boat is traveling at a velocity of 15 m/s to the east (@ 0o).

b. Southward velocity?

The boat is traveling at a velocity of 10 m/s to the south (@ 270o).

c. Resultant velocity?

The boat is traveling at a velocity of 18 m/s to the southeast (@ 326o).

9. A plane flying due North at 100 m/s, is blown due West at 50 m/s by a strong wind. What is the plane's resultant velocity?

The resultant velocity of the plane is 110 m/s @ 117o.

10. A motor boat heads due East at 16 m/s across a river that flows due South at 9.0 m/s.

a. What is the resultant velocity of the boat?

18 m/s @ 331o

b. How long does it take the boat to cross the 136 m wide river?

The amount of time to cross depends on the width of the river (east-west distance) and the velocity of the boat to the east).

c. How far downstream is the boat when it reaches the other side of the river?

The distance the boat travels downstream depends on the velocity of the boat to the south (due to current) and the amount of time in the water.

11. What is meant when physicists say that perpendicular vectors are independent.

Perpendicular vectors are independent" refers to the fact that a quantity that acts in a certain direction has no effect on another quantity that acts in a direction that is 90o with respect to the first.

For example, a force (vector quantity) that pushes to the west on a stationary object will cause that object to accelerate to the west. The object will travel faster and faster to the west. However, the object will not accelerate to the north or south since the force is perpendicular to these directions.

 

A force that pushes to the north on an object that is moving to the west at a constant speed will cause the object to accelerate (gain speed) to the north. However the car will maintain its constant velocity to the west since a vector to the north will not affect a vector to the west.

12. Two forces, 110 N at Oo and 55 N at 90o, act on a point. What is the resultant force?

Resultant force is 120 N @ 27o.

13. A motorboat heads due East across a river (41 meters wide). The boat can move at 3.8 m/s with respect to the water. The current is flowing South at 2.2 m/s.

a. What is the boat's velocity as seen by an observer on shore?

4.4 m/s

b. How much time does it take for the boat to cross the river?

10.8 s

c. How far downstream is the boat when it reaches the other side?

23.6 m

14. A boat travels at 3.8 m/s, South with respect to the water and heads straight across a river 240 meters wide. The river flows at 1.6 m/s, East.

a. What is the boat's resultant speed with respect to the river bank?

4.1m/s

b. How long does it take the boat to cross the river?

63s

c. How far downstream is the boat when it reaches the other side?

101m

15. A river flows due South. A riverboat pilot heads the boat 27o and is able to go straight across the river with a resultant velocity of 6.0 m/s.

a. What is the velocity of the current?

b. What is the velocity of the boat?

16. Digit Tal wishes to throw a ball at an angle of 40 degrees. He calculates that he can achieve this if he throws the ball with a vertical velocity of 20 m/s and a horizontal velocity of 23.84 m/s.

a. What is his resultant velocity?

b. Was his calculation correct? Is the angle 40 degrees?

17. Mr. Schmidgall throws a ball with a velocity of 45 m/s (convert this to miles per hour before you try and get a hit off his pitching). Determine the horizontal and vertical components of this velocity if he throws the ball at:

a. 30 degrees.

b. 45 degrees.

c. 60 degrees.

d. 90 degrees.

18. How much will the horizontal velocities of each of the above throws affect the vertical velocities? Explain.

The horizontal velocities will not affect the vertical velocities since perpendicular vector quantities do not affect each other.

19. A cat is dropped (accidentally of course) from the top of a building that is 25 meters high.

a. What is the initial vertical velocity of the cat?

Vyo=0m/s

b. Determine the time it takes the poor cat to reach the pavem...OK safety net 25 meters below.

c. Determine the velocity of the cat as she strikes the safety net.

20. The cat survives, scratches her owner and is throw horizontally with a velocity of 15 m/s (not so accidentally) from the top of a building that is 25 meters high.

a. What is the initial vertical velocity of the cat?

If thrown horizontally, the initital vertical velocity is 0 m/s.

b. Determine the time it takes the poor cat to reach the pavem...OK safety net 25 meters below.

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c. Determine how far the cat will travel horizontally by the time it has fallen 25 meters (again to place the safety net).

21. A stone is thrown horizontally at a speed of 10.0 m/s from the top of a cliff 80 meters high.

a. How long does it take the stone to reach the bottom of the cliff?

b. How far from the base of the cliff does the stone strike the ground?

22. Harry the human cannon ball is shot horizontally at 20.0 m/s from the top of a 49.0 meter high tower. How far from the base of the tower should the safety net be placed for Harry?

23. Sadan Alone is standing on the top of a cliff. Her x-boyfriend is standing 47 meters from the base of that same cliff. Being a physicist, Sadan calculates that she needs to hurl a water balloon horizontally with a velocity of 15 m/s to hit her boyfriend squarely on top of his head. How high is she above the crown of her x-boyfriend's head?

24. Lepo Faith stands on top of a relatively high hill. He gets a running start, and becomes airborne with a horizontal velocity of 13 m/s. If Lepo strikes the ground a horizontal distance of 15.0 meters from his take off point, how tall is the hill from which he jumped?

25. An arrow is fired horizontally with a speed of 89 m/s directly at the bullseye of a target 60 meters away. When it is fired, the arrow is 1.0 meters above the ground. How far short of the target does it strike the ground? (Careful. This is not just asking you to find the horizontal distance it will travel by the time the arrow hits the ground.)

26. Use your work from challenge 17 to determine the time in the air for each of the four throws.

The total time in the air depends on the vertical component of the velocity. The time to reach the peak can be found using:

The time to reach the peak is the same as the time to come back down

a.

b.

c.

d.

27. Omiachen Head, an archer, is riding in the back of a pickup truck which is traveling at a constant velocity of 40 miles per hour. He decides to fire off his last arrow by shooting it straight up into the air. He reasons to himself that if he shoots the arrow straight up, it will come down at the same location from which it was shot. This way he would be far away from the arrow when it came down. After all, he's moving horizontally at 40 miles an hour. He'll be far away. Right? Explain.

Since the arrow is shot from the back of a truck which is traveling at a velocity of 40 mi/hr, the arrow also has a horizontal velocity of 40 mi/hr. Once launched, the arrow will also have a vertical component of its velocity, but its horizontal component will remain at 40 mi/hr (unless air resistance comes into play). As a result, the horizontal displacement of the arrow will be the same as the truck since they are traveling at the same speed, in the same direction for the same amount of time...BONK!

28. A projectile is fired at such an angle from the horizontal that the vertical component of its velocity is 49 m/s. The horizontal component of its velocity is 61 m/s.

a. How long does it take the projectile to rise to its highest point (where its vertical velocity is zero)?

b. How long does the projectile remain in the air?

tup = tdown, ttotal = tup + tdown= 5.0 s + 5.0 s = 10.0 s

c. What horizontal distance does it travel during its trip (the projectile's range)?

d. What is the initial velocity of the projectile?

29. Harry the human cannonball is now fired with a velocity of 30 m/s at an angle of 60o with the horizontal. Determine Harry's:

a. initial vertical and horizontal velocity components.

b. total time in the air.

c. horizontal distance traveled (where the net is hopefully located).

d. maximum height achieved (which is hopefully lower than the ceiling).

30. A projectile is fired at an angle of 53o with respect to the horizontal. The speed of the projectile is 200 m/s. Determine the:

a. total time the projectile is in the air.

33s

b. range the projectile travels by the time it comes down.

3960m

31. Omiachen Head, an archer, is riding in the back of a pickup truck which is traveling at a constant velocity of 22 m/s. He decides to fire off his last arrow by shooting it straight up into the air. The truck is moving a constant velocity of 22 m/s, and Omiachen fires the arrow straight up with a vertical velocity of 98 m/s.

a. How long does the arrow remain in the air by the time it returns to its firing height.

20s

b. What horizontal distance does the arrow travel while it is in the air?

440m

c. How does the horizontal distance the arrow travels while in the air compare the the horizontal distance Omiachen travels?

Since Omiachen and the truck have the same horizontal velocities, which remain constant, their horizontal displacements will be the same also.

32. A golf ball is hit at an angle of 45o with respect to the horizontal. If the initial velocity of the ball is 52 m/s, how far will it travel horizontally before striking the ground?

33. A toy car moves off the edge of a table that is 1.5 meters high. If the car lands 0.5 meters from the base of the table,

a. how long did it take for the car to fall to the floor?

b. with what horizontal velocity was the car moving?

34. Divers at Acapulco dive from a cliff that is 61 meters high. If the rocks below the cliff extend outward for 23 meters, what is the minimum horizontal velocity a diver must have to clear the rocks?

35. A heavy box is pulled across a wooden floor with a rope. The rope makes an angle of 60o with the floor. A force of 80 Newtons is exerted on the rope. What is the component of the force parallel to the floor.

36. A lawn mower is pushed by applying a force of 72 Newtons along the handle. Find the force that pushes it across the lawn (horizontal component) when the handle is held at an angel of ___ with respect to the lawn.

A. 60o B. 40o C. 30o

37. Two forces of 50 Newtons and 80 Newtons act concurrently on one object.

a. Determine the minimum resultant force that can be achieved with these two forces.

80 N - 50 N = 30 N

b. Determine the maximum resultant force that can be achieved with these two forces.

80 N + 50 N = 130 N

38. Ben Lifting pushes Ima Wimp to the East with a force of 100 N. You, being a good Samaritan, decide to balance the force placed on Ima by Ben. What force do you apply to Ima and in what direction to keep him in equilibrium.

To create equilibrium, you must push to the west with a force of 100 N.

39. For each of the following choices explain why it is correct or incorrect.

If an object is in equilibrium it must:

a. not be moving.

An object in equilibrium may not be moving or it may be moving at a constant velocity: incorrect.

b. not be accelerating.

An object in equilibrium may not be moving or it may be moving at a constant velocity. In either case, it is not accelerating: correct.

c. be accelerating and moving.

An object in equilibrium may not be moving or it may be moving at a constant velocity. In either case, it is not accelerating: incorrect.

40. Two forces act upon an object. One force is 6.0 N, North while a second is 8.0 N, East.

a. Determine the magnitude and direction of the resultant of these two forces.

b. Determine the magnitude and direction of the force that produces equilibrium.

Equilibrium will be produced by a force of the same magnitude as the resultant but in the opposite direction (directions differ by 180o): 10 N @ 217o.

41. A 36 N force acts at 180o while a 48 N force acts at 120o.

a. What is the resultant force of the 36 N and 48 N forces?

b. What force would balance these two forces and keep the system in equilibrium?

73 N @ 305o

42. A sign is held up by two strings. Each string produces a vertical force equal to half the weight of the sign. If the sign has a weight of 30 Newton, determine the tension on each string if the angle between the strings is

a. 70o b. 90o c. 130o

 

43. A trunk that weighs 500 N is placed on an inclined plane that forms a 66o with the horizontal.

a. What is the value of the force that pushes the trunk into the plane (F*)?

b. What is the force that pushes the trunk down along the inclined plane (FII)?

c. As the angle of the plane increases, which component of weight increases and which component decreases? Explain.

As the angle increases the parallel component increases since the angle between it and the weight vector decreases (Fgrav and F// are closer to being in the same direction.

44. A 1200 kg car is parked on a 36o incline.

a. Find the force that tends to cause the car to accelerate down the hill.

b. Find the force that pushes the car down into the hill.

45. The American Eagle at Great America has carts with masses of 725 kg. If the first drop is declined at an angle of 35o:

a. Determine the normal force acting upon the coaster car.

b. Determine the net forces acting parallel to and perpendicular to the track.

c. Determine the parallel and perpendicular acceleration of the coaster car.

46. On September 11, 1972 Kent Wegley (Mr. Wegley's older but not wiser brother) challenged Devil's Hill in Roy Utah on his newly mastered bicycle. Kent had a mass of 28 kg and Devil's Hill was declined at an angle of 40o.

a. Determine the normal force acting upon Kent while he was accelerating down Devil's Hill.

210 N

b. Determine the net forces acting parallel to and perpendicular to Devil's Hill.

180 N

c. Determine Kent's parallel and perpendicular acceleration.

6.3 m/s2

Note: The above event is true. The name has not been changed to protect the not so innocent and an attempt to take a sharp turn at a high speed on a bike that had no brakes resulted in Kent's collision with a parked car, which transformed him into a projectile. All this left him with legendary status with our friends and a broken nose.

 

47. A runner runs with a velocity of 22 miles/hour, 150o.

a. How fast is she running North?

22sin150o=11m/s

b. How fast is she running West?

-(22cos150o)=19.1m/s

48. A street lamp weights 150 N. It is supported equally by two wires that form an angle of 120o with each other. What is the tension on each of the wires?

49. A stone is thrown horizontally at 8.0 m/s the top of a building 20 meters high.

a. How long will it take the stone to reach the ground?

b. How far from the base of the building will the stone land?

d=(2.02s)(8m/s)=16.16m


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Last Updated: October 8, 1999