1'4PROJECTILE MOTION

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1.4 PROJECTILE MOTION
1.4.1 Independence of Horizontal and Vertical
Motions for Projectiles
1.4.2 Terminal Velocity
1.4.3 Real Trajectories of Projectiles
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1.4.1 Independence of Horizontal and Vertical
Motions for Projectiles

• A stroboscopic photograph above shows the the
  motion of two balls, one released from rest and
  the other projected simultaneously with a
  horizontal velocity.

• The vertical displacements of the two balls at
  each stroboscopic interval are . This implies
  that the vertical motion of a projectile is
  its horizontal motion.
• The horizontal displacement of the first ball is
  equal in each equal time interval, which implies
  that the motion along the horizontal
  direction is the vertical acceleration
  of the projectile.

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• Mathematical Treatment for Projectile Motion
• Consider a body projected obliquely from O with
  the velocity u at an angle ? to the horizontal as
  shown. H and R respectively represent the
  maximum height attained and horizontal range of
  the body.
• Vertical Motion (y-direction)
• Vertical acceleration ay ,
• Vertical initial velocity uy ,
• Vertical displacement at time t
• y(t) . Eq.(1)

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• Horizontal Motion (x-direction)
• Horizontal initial velocity ux ,
• Horizontal displacement at time t
• x(t) . Eq.(2)
• Equation of trajectory
• Eliminating the parameter in Eqs.(1) and
  (2), we get
• y , Eq.(3)
• which is a passing through the .
• Time of flight (T)
• The total time taken for the whole journey is
  found by putting y(T) in Eq.(1),
• T Eq.(4)

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• Maximum height attained (H)
• At the maximum height, the equals zero, and
  so by equation of motion,
• H Eq.(5)
• Horizontal range (R)
• The horizontal range is obtained by putting t
  in Eq.(2), that is
• R Eq.(6)

Simulating projectile motion
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• Trajectories for different projection angle
  (constant u)
• - What are the angles of the trajectories in
  dotted lines?
• - If the object is projected above the ground
  level instead, would the projection angle for
  optimum range still be 45o?

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• Monkey and Hunter Experiment
• The following figure shows the set up for the so
  called monkey-and-hunter' experiment, where the
  gun is aiming at the target. At the instant the
  bullet emerges from the gun, it breaks the
  contact and allows the electromagnet to release
  the monkey'. The question is can the bullet
  still hit the monkey?

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• Height of monkey at time t hm(t) .
• Height of bullet at time t hb(t) .
• Time for the bullet to reach the monkey T .
• Height of monkey at T hm(T) .
• Height of bullet at T hb(T) .

Can you verify this
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Examination questions

• 1990-IIA-4
• A hunter aims his gun at a target which is at
  rest at point M, and his gun makes an angle ?
  with the horizontal. Exactly as the gun is fired
  the target drops from M with zero initial
  velocity. If the bullet is to strike the target,
  the angle ? depends on
• (1) u, the initial speed of the bullet.
• (2) h, the vertical height of the target above
  the level of the gun.
• (3) d, the horizontal distance of the gun from
  the target.
• A. (1), (2) and (3) B. (1) and (2) only C.
  (2) and (3) only
• D. (1) only E. (3) only

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• 1993-IIA-4
• Two identical coins P and Q are placed at the
  edge of a table. At the same instant, P is
  pushed slightly and falls vertically to the
  ground while Q is projected horizontally and
  reaches the ground through a parabolic path (as
  shown). Which of the following statements is/are
  correct? (Neglect air resistance.)

(1) P and Q reach the ground at the same
time. (2) P and Q have the same
acceleration. (3) P and Q have the same
vertical speed on reaching the ground. A. (1)
only B. (3) only C. (1) and (2)
only D. (2) and (3) only E. (1), (2) and (3)
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• 1994-IIA-6
• Three bombs are released from a bomber flying
  horizontally with constant velocity to the right.
  They are released from rest (relative to the
  bomber) one by one at one-second intervals.
  Neglecting air resistance, which of the following
  diagrams correctly shows the positions of the
  bomber and the three bombs at a certain instant?
• A. B. C.
• D. E.

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• 1996-IIA-3
• A particle is projected horizontally from a
  table with an initial speed u and attains a speed
  v just before hitting the ground. What is the
  time of flight of the particle? (Neglect air
  resistance.)
• A. B. C. D.
• E. It cannot be found as the vertical distance
  fallen is not known.
• 1998-IIA-6
• A small object is thrown horizontally towards a
  vertical wall 1.2 m away. It hits the wall 0.8 m
  below its initial horizontal level. At what
  speed does the object hit the wall? (Neglect air
  resistance.)
• A. 2 ms-1 B. 3 ms-1 C. 4 ms-1
• D. 5 ms-1 E. 7 ms-1

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• 2001-IIA-3
• A stone is projected at an angle of 45o to the
  horizontal with an initial kinetic energy E.
  Neglecting air resistance, when the stone is
  halfway up, its kinetic energy is
• A. B. C. D. E.
• 2004-IIA-3
• The figure shows the barrel of a gun that aims
  directly at a point P 40 m from the muzzle of the
  gun. The barrel makes an angle q with the
  vertical. If the speed of the bullet is 50 m s-1
  when it leaves the gun, calculate the separation
  between the bullet and point P when the bullet is
  vertically below P. (Neglect air resistance.)
• A. 3.2 m
• B. 4.8 m
• C. 7.8 m
• D. It cannot be found as the
• value of q is not known.

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1.4.2 Terminal Velocity
In the absence of resistive (or dragging) force,
all objects released from the same height should
fall with acceleration. In the presence of
resistive force, however, an object released from
a certain height should fall with
acceleration. Furthermore, heavier objects fall
than lighter objects in real situation
and it takes time for objects to fall in
liquid columns than in air columns.

• Dragging force
• For a body falling in a fluid, it would be
  assumed that the dragging force is directly
  proportional to the of the body and in the
  direction to the motion of the body, i.e.,
• Eq.(12)

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• Equation of motion
• In the following, the point would be taken as
  the origin, and direction would be taken as
  positive.
• The net force acting on the falling body is
• F
• and so according to Newtons second law, the
  acceleration is
• a
• which varies according to v.
• From Eq.(4), we get
• Eq.(13)

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• Terminal speed
• At the terminal speed vT, the dragging force has
  grown to a size that the weight of the body,
  and hence
• which gives the size of terminal speed
• vT Eq.(14)
• The terminal speed is higher for massive
  objects.
• The constant b is called the , which
  increases with both the of the objects
  and the of the medium through which the
  objects fall.

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• Exact variation of v with t
• According to Eq.(13),

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• Examination questions
• 1987-IIA-2
• A parachutist of mass m falls in air under the
  influence of gravity. The air resistance is
  equal to bv, where v is his speed and b is a
  constant. After falling a height s from rest, he
  reaches a terminal speed u. His kinetic energy
  at that instant is
• A. mgs.
• B. mga bus.
• C. mgs (m3g2)/(2b2).
• D. mgs (m3g2)/(2b2).
• E. m3g2/2b2.

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• 1996-IIA-4
• An object is thrown vertically upward and
  experiences an air resistance opposing its motion
  with magnitude proportional to its speed. Which
  of the following graphs best represents the
  variation of the acceleration, a, of the object
  with time, t, starting from the moment when the
  object leaves ones hand up to the time when it
  returns to the ground?
• A. B.
• C. D.
• E.

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1.4.3 Real Trajectories of Projectiles
By ideal trajectory', we mean the trajectory for
objects with different masses, shapes and sizes
are the same for given values of and .
In reality, however, both the maximum height
and horizontal range are much by air
resistance. Moreover, the actual trajectories
are , i.e., the projectiles come down more
steeply than it rises.