Pavel Panchekha


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Problem Set 1 (Given 9/18/11)

Level I

  1. Give the \(x\) and \(y\) components of the vector of magnitude 15 directed 135 degrees counterclockwise from the positive $x$-axis.
  2. Graph and find the sum of the following three vectors in component form and give the magnitude and direction: ⟨+4, +2⟩, ⟨-7, 2⟩, and the vector of magnitude 8 directed 30 degrees counterclockwise from the positive $x$-axis.
  3. Find the angle between the following two vectors: ⟨+9,+4,-3⟩ and ⟨-5,-2,+7⟩

Level II

  1. One can derive a kinematics equation describing the motion of an object with constant acceleration that contains no acceleration term. Starting with the equations you know, find this equation in the form with the change in displacement on the left hand side.
  2. Superman jumps from a building and flies in a straight line at speed vs. At \(t=0\), he sees the batcave straight ahead of him at a ground distance of \(D\) meters. At the same time, the batmobile leaves the batcave, starting from rest and accelerating with constant acceleration a in the same direction. For what values of a will superman and the batmobile have the same ground position twice? Once? Never?
  3. At \(t=0\), a man stands 5000 meters below a bird, which in turn is flying 6000 meters below a plane. The bird flies at 10 m/s and the plane flies at 100 m/s. Both the plane and the bird fly parallel to the ground (assumed to be flat), but the bird flies at an angle of 20 degrees north of east, while the plane flies at 20 degrees east of north. Find an equation describing the position of the bird in the reference frame of each of the three observers.

Level III

  1. A pebble is lodged in the rim of a tire of radius \(r\) rolling forward without slipping. The center of the tire moves in a straight line with speed \(v\). Suppose that at \(t=0\), the pebble lies at the origin directly under the center of the tire. Find the position of the pebble at an arbitrary time later.
  2. A hiker wants to lift a package of food to keep it out of reach of bears. She tosses a rope of length \(l\) over a tree such that one end rests tied to the package on the ground and the other in her hand at height \(h\) (\(h < (1/2)l\)). She then walks back at constant speed \(v\) holding the rope (starting directly under the tree at \(t=0\)). Find the height of the package as a function of time.