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Horizontal Projectile

A body be projected horizontally from a certain height y vertically above the ground with initial velocity u (Fig. 8). If friction is considered to be absent, the horizontal velocity remains constant.
 
31538.png
Fig. 8

Trajectory of horizontal projectile

The horizontal displacement x is governed by the equation
 
30450.png
The vertical displacement y is governed by
 
30444.png
(since initial vertical velocity is zero)
 
By substituting the value of t in (5), 30438.png.

Displacement of projectile

After time t, horizontal displacement x = ut and vertical displacement 30426.png.
 
So, the position vector 30420.png
∴​  30413.png
and 30406.png 30400.png30394.png

Instantaneous velocity

Throughout the motion, the horizontal component of the velocity is vx = u.
 
The vertical component of velocity increases with time and is given by
vy = 0 + gt = gt  (From v = u + gt)
 
So, 30388.png = 30382.png
i.e., 30376.png 30370.png
 
Again 30364.png
i.e., 30358.png

Direction of instantaneous velocity

30352.png
 30346.png 30339.png30333.png
where φ is the angle of instantaneous velocity from the horizontal.

Time of flight

If a body is projected horizontally from a height h with velocity u and time taken by the body to reach the ground is T, then
 
30327.png

Horizontal range

Let R is the horizontal distance traveled by a body, as shown in Fig. 9.
 
30315.png
Fig. 9
 
30309.png
 
If projectiles A and B are projected horizontally with different initial velocities from same height and third particle C is dropped from same point then
  • All three particles will take equal time to reach the ground.
  • Their net velocity would be different but all three particles possess same vertical component of velocity.
     
    If various particles thrown with same initial velocity but indifferent direction (Fig. 10), then
30297.png
Fig. 10
  • They strike the ground with same speed at different times irrespective of their initial direction of velocities.
  • Time would be least for particle E which was thrown vertically downward.
  • Time would be maximum for particle A which was thrown vertically upward.




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