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Projectile Motion

Projectile A body which is in flight through the atmosphere but is not being propelled by any fuel is called projectile.

Assumptions of projectile motion

  • There is no resistance due to air.
  • The effect due to curvature of earth is negligible.
  • The effect due to rotation of earth is negligible.
  • For all points of the trajectory, the acceleration due to gravity g is constant in magnitude and direction.

Principles of physical independence of motions

  • The motion of a projectile is a two-dimensional motion. So, it can be discussed in two parts: horizontal motion and vertical motion. These two motions take place independent of each other. This is called the principle of physical independence of motions.
  • The velocity of the particle can be resolved into two mutually perpendicular components. Horizontal component and vertical component.
  • The horizontal component remains unchanged throughout the flight. The force of gravity continuously affects the vertical component.
  • The horizontal motion is a uniform motion and the vertical motion is a uniformly accelerated retarded motion.
  • In projectile motion, the horizontal component of velocity (u cos θ), acceleration (g) and mechanical energy remain constant, while, speed, velocity, vertical component of velocity (u sin θ), momentum, kinetic energy and potential energy, all change.
  • Velocity and KE are maximum at the point of projection while minimum (but not zero) at highest point.

Equation of trajectory

A projectile is thrown with velocity u at an angle θ with v cos θ component along X-axis and u sin θ component along Y-axis (Fig. 1).
 
30973.png
Fig. 1
 
30967.png   (1)
Equation (1) shows that the trajectory of projectile is parabolic because it is similar to equation of parabola,
 
y = ax – bx2
 
Equation of oblique projectile also can be written as:
 
30961.png
 
(where R = horizontal range = 30955.png)

Displacement of projectile

Let the particle acquires a position P having the coordinates (x, y) just after time t from the instant of projection. The corresponding position vector of the particle at time t is 30943.png as shown in Fig. 2.
 
30937.png
Fig. 2
 
30931.png (2)
 
Putting the values of x and y in (2) we obtain the position vector at any time t as
 
30925.png
⇒ 30918.png
and 30912.png
30906.png
or 30900.png
 
30894.png
Fig. 3

 

Note: The angle of elevation φ of the highest point of the projectile and the angle of projection θ are related to each other as
31206.png

Instantaneous velocity v

In projectile motion, the vertical component of velocity changes but horizontal component of velocity remains always constant.
 
Let vi be the instantaneous velocity of projectile. At time t, direction of this velocity is along the tangent to the trajectory at point P.
 
30863.png
30857.png
30851.png
30845.png
 
Direction of instantaneous velocity,
30838.png
or 30832.png

Change in velocity

  • Between projection point and highest point:
     
    30826.png
  • Between complete projectile motion:
     
    30818.png

Change in momentum

  • Between projection point and highest point:
     
    30812.png
  • For the complete projectile motion:
     
    30806.png

Angular momentum

The angular momentum of projectile at highest point of trajectory about the point of projection is given by
 
L = mvr 30800.png
 
∴ 30794.png

Time of flight

The total time taken by the projectile to go up and come down to the same level from which it was projected is called time of flight.
 
Time of flight, 30788.png 
  • Time of flight can also be expressed as: 30782.png (where uy is the vertical component of initial velocity).
  • For complementary angles of projection θ and 90° – θ,
Ratio of time of flight
30776.png
= tan θ  30770.png
 
Multiplication of time of flight
30764.png 
⇒ 30758.png
  • If t1 is the time taken by projectile to rise upto point P and t2 is the time taken in falling from point P to ground level (Fig. 4), then
     
    30752.png time of flight
30745.png
Fig. 4
 
or 30739.png 
and height of the point P is given by
30733.png 30727.png 
By solving 30721.png
  • If B and C are at the same level on trajectory and the time difference between these two points is t1, similarly A and D are also at the same level and the time difference between these two positions is t2 (Fig. 5), then
31510.png
Fig. 5
 
30715.png

Horizontal range

It is the horizontal distance traveled by a body during the time of flight (Fig. 6).
 
30703.png
Fig. 6
 
30697.png
  • Range of projectile can also be expressed as:
     
    R = u cosθ × T
     
    30691.png
     
    (where ux and uy are the horizontal and vertical components of initial velocity.)
  • If the angle of projection is changed from θ to θ’ = (90° – θ) then range remains unchanged.
     
    30679.png
So a projectile has same range at angles of projection θ and (90° – θ), though time of flight, maximum height, and trajectories are different.
 
These angles θ and 90° – θ are called complementary angles of projection and for complementary angles of projection, ratio of range
 
30673.png
 30666.png
  • For angle of projection θ1 = (45° – α) and θ2 = (45° + α), range will be same and equal to u2 cos 2α/gθ1 and θ2 are also the complementary angles.

Maximum range

A projectile will have maximum range when it is projected at an angle of 45° to the horizontal and the maximum range will be (u2/g).
 
When the range is maximum, the height H reached by the projectile
30660.png
 
i.e., if a person can throw a projectile to a maximum distance Rmax, the maximum height to which it will rise is (Rmax/4).
  • Relation between horizontal range and maximum height:
     
    30648.png and 30642.png
     
    ∴ 30636.png
     
    ⇒ R = 4 H cot θ
  • If in case of projectile motion range R is n times the maximum height H,
     
    i.e., R = nH
     
    ⇒ 30629.png  tan θ = [4/n]
     
    or θ = tan–1 [4/n]
     
    The angle of projection is given by θ = tan–1 [4/n].
Note:
If R = H, then θ = tan–1(4) or θ = 76° or θ = 76°.
 
If R = 4H, then θ = tan–1(1) or θ = 45°.

Maximum height

It is the maximum height from the point of projection a projectile can reach.
 
So, by using v2 = u2 + 2as
 
0 = (u sin θ)2 – 2gH
 
30598.png
  • Maximum height can also be expressed as:
     
    30590.png (where uy is the vertical component of initial velocity).
     
    30584.png (when sin2θ = max = 1, i.e., θ = 90°)
     
    i.e., for maximum height, body should be projected vertically upward. So it falls back to the point of projection after reaching the maximum height.
  • For complementary angles of projection θ and 90° – θ,
     
    Ratio of maximum height
     
    30578.png 30572.png = tan2 θ
     
    ∴ 30565.png

Projectile passing through two different points on same height at time t1 and t2

If a particle passes two points situated at equal height y at t = t1 and t = t2 (Fig. 7), then
 
30559.png
Fig. 7
  • Height (y): 30553.png  (3)
     
    and 30547.png  (4)
Comparing (3) with (4), we get
 
30541.png
 
Substituting this value in equation (3), we get
 
30535.png  30529.png
  • Time (t1 and t2): 30523.png
     
    30517.png
     
    ⇒ 30511.png
     
        30505.png
     
    and 30499.png

Motion of a projectile as observed from another projectile

The motion of a projectile relative to another projectile is a straight line.

Energy of projectile

When a projectile moves upward, its kinetic energy decreases, potential energy increases, but the total energy always remains constant.
 
If a body is projected with initial kinetic energy K[ = (1/2)mu2], with angle of projection θ with the horizontal then at the highest point of trajectory
  • Kinetic energy 30492.png
     
    ∴ K’ = K cos2 θ
  • Potential energy = mgH
     
    30486.png32004.png
     
    30474.png
  • Total energy = Kinetic energy + Potential energy
     
    30468.png
     
    30462.png
     
    = Energy at the point of projection.
This is in accordance with the law of conservation of energy.




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