Advanced Problems 1

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Advanced Problems 1

• These problems will contain
• Basic concepts of velocity and acceleration.
• Usage of proper physics 1 kinematics equations.
• Some calculus techniques, such as derivatives and
  integration.

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• An indestructible bullet 2cm long is fired
  straight through a board that is 10cm thick. The
  bullet strikes the board with a speed of 420m/s
  and emerges with a speed of 280m/s.
• What is the average acceleration of the bullet as
  it passes through the board?

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• What is the total time that the bullet is in
  contact with the board?

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• What thickness of board would it take to stop the
  bullet (assuming the bullets acceleration though
  the board remains the same)

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• 2. A student throws a set of keys vertically
  upward to her sorority sister, who is in a window
  4m above. The keys are caught 1.5s later by the
  sisters outstretched hand.
• With what initial velocity were the keys thrown?

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• What was the velocity of the keys just before
  they were caught?

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• 3. The height of a helicopter above the ground is
  given by h 3t3, where h is in meters and t is
  in seconds. After 2 seconds, the helicopter
  releases a small mailbag. How long after its
  release does the mailbag reach the ground?

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• Automotive engineers refer to the time rate of
  change of acceleration as the jerk. If an
  object moves in one dimension such that its jerk
  J is constant,
• Determine expressions for its acceleration,
  velocity, and position given that its initial
  acceleration, velocity and speed are ai, vi, and
  xi respectively.

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• b. Show that

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• 5. The speed of a bullet as it travels down a
  barrel of a rifle toward the opening is given by
  v -5x107t2 3x105t. The acceleration of the
  bullet as it just leaves the barrel is zero.
• Determine the acceleration and position of the
  bullet as a function of time when the bullet is
  in the barrel.

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• Determine the length of time the bullet is
  accelerated.
• c. Find the speed at which the bullet leaves the
  barrel.
• d. What is the length of the barrel?

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• A daring ranch hand sitting on a tree limb wishes
  to drop vertically onto a horse galloping under a
  tree. The speed of the horse is 10m/s, and the
  distance from the limb to saddle is 3m.
• What must be the horizontal distance between the
  limb and the saddle when the ranch hand drops?

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• A test rocket is fired vertically upward from a
  well. A catapult gives it an initial velocity of
  80m/s at ground level. Subsequently, its engines
  fire and it accelerates upward at 4m/s2 until it
  reaches an altitude of 1000m. At that point the
  engines fail, and the rocket goes into free fall.
• How long is the rocket in motion above the
  ground?
• What is its maximum altitude?
• What is its velocity just before it hits the
  ground?

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• 8. A physics student and a mountain climber
  climbs a 50m cliff that overhangs a calm pool of
  water. He throws two stones vertically downward,
  1second apart, and observes that they cause a
  single splash. The first stone has an initial
  speed of 2m/s.
• How long after the release of the first stone do
  the stones hit the water?
• What was the initial velocity of the second
  stone?
• What is the velocity of each stone at the instant
  the two hit the water?

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9. You are standing on the ground at the origin
of a coordinate system. An airplane flies over
you with constant velocity parallel to the x axis
and at a constant height of 7.6x103m. At t0, the
plane is directly above you. At t30s, the
position vector from you to it is given by
P30(8.04x103m)i (7.6x103m)j. Determine the
magnitude and orientation of the planes position
at t45s.
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• Given the displacement vectors
• A (3i-4j4k)m and B (2i3j-7k)m find the
  magnitudes of C A B and D 2A B also
  expressing each in terms of its x, y, and z
  components.

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• A radar station locates a sinking ship at range
  17.3km and bearing 136 clockwise from north.
  From the same station a rescue plane is at
  horizontal range 19.6km 153 clockwise from
  north. With elevation 2.2 km.
• Write the vector displacement from plane to ship
  letting i represent east, j represent north, and
  k represent up.
• How far apart are the plane and ship?

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• A vector is given by R2i 1j 3k.
• a. Find the magnitudes of the x, y, and z
  components.
• b. Find the magnitude of R.
• c. Find the angles between R and the x, y, and z
  axes.

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13. In general, the instantaneous position of an
object is specified by its position P leading
from a fixed origin to the location of the
object. Suppose that P 4i3j-2tj where P is in
meters and t is in seconds what is the derivative
of this position function? What does this
derivative represent about the object?
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14. A jet airliner, moving initially at 300mi/h
to the east, suddenly enters a region where the
wind is blowing at 100mi/h in a direction 30
north of east. What are the new speed and
direction of the aircraft relative to the ground?
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• A fish swimming in a horizontal plane has
  velocity vi (4i 1j)m/s at a point in the
  ocean whose displacement from a certain rock is
  ri (10i 4j)m. After the fish swims with
  constant acceleration for 20s, its velocity is v
  (20i 5j)m/s
• What are the components of the acceleration?
• What is the direction of the acceleration with
  respect to the unit vector i?
• Where is the fish at t25s if it maintains its
  original acceleration and in what direction is it
  moving?

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• A particle initially located at the origin has an
  acceleration of a 3j m/s2 and an initial
  velocity of vi 5i m/s.
• Find the vector position and velocity at any time
  t.
• Find the coordinates and speed of the particle at
  t2s.

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• 17. A projectile is fired in such a way that its
  horizontal range is equal to 3 times its maximum
  height. What is the angle of projection?

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• 18. A placekicker must kick a football from a
  point 36m from the goal, and half the crowd hopes
  it will clear the crossbar, which is 3.05m high.
  When kicked, the ball leaves the ground with a
  speed of 20m/s at an angle of 53 to the
  horizontal.
• By how much does the ball clear or fall short of
  the crossbar?
• Does the ball approach the crossbar while still
  rising or while falling?

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• 19. A train slows down as it rounds a sharp
  horizontal curve, slowing from 90km/h to 50km/h
  in the 15 seconds that it takes to round the
  curve. The radius of the curve is 150m. Compute
  the acceleration at the moment the train speed
  reaches 50km/h. Assume that the train slows down
  at a uniform rate during the 15 second interval.

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• The determined coyote is out once more to capture
  the roadrunner. The coyote wears Acme jet powered
  roller skates which provide constant horizontal
  acceleration of 15m/s2. The coyote starts off at
  rest 70m from the edge of a cliff the instant
  that the roadrunner zips past him in the
  direction of the cliff.
• If the roadrunner moves with constant speed,
  determine the minimum speed he must have to reach
  the cliff before the coyote.
• If the cliff is 100m above the floor of the
  canyon, determine where the coyote lands after
  missing the roadrunner (his skates still thrust
  while in midair).
• c. What is his impact velocity?