Motion in a plane

6. Projectile Motion

Basic Concept

1. Projectile motion is a form of motion in which an object or particle (called a projectile) is thrown with some initial velocity near the earth's surface, and it moves along a curved path under the action of gravity alone.
2. It is an example of two dimensional motion with constant acceleration.
3. If $F$ is constant then $a$ is constant and when force is oblique direction with initial velocity , the result is parabolic path.
4. Projectile motion is considered as two simultaneous motion in mutually perpendicular directions which are completely independent from each other i.e., horizontal motion and vertical motion

Projectile thrown at an angle with horizontal

So we have seen our basics let us now learn the projectile motion in little more detail about Projectile thrown at an angle with horizontal. There are certain approximate assumptions we make while studying projectile motion and they are
1. We ignore the frictional resistance due to air.
2. The effect due to rotation of earth and the curvature of earth is negligible.
3. The acceleration due to gravity is constant in magnitude and direction at all points during the motion of projectile.
• Projectile motion is a special case of motion in two dimension when acceleration of particle is constant in both magnitude and direction.
• An object is referred as projectile when it is given an initial velocity which subsequently follows a path determined by gravitational forces acting on it. For example bullet fired from the rifle javelin thrown by the athlete etc.
• Path followed by a projectile is called its trajectory.
• In this section we will study the motion of projectile near the earth surface. We would be neglecting the air resistance.
• Acceleration acting on a projectile is constant which is acceleration due to gravity (g=9.81 m/s) directed along vertically downward direction.
• We shall treat the projectile motion in a Cartesian co-ordinates system taking y axis in vertically upwards direction and x axis along horizontal directions
• Now x and y components of acceleration of projectile is
${a_x} = 0$ and ${a_x} = 0$
Since acceleration in horizontal direction is zero, this shows that horizontal component of velocity is constant and vertical motion is simply a case of motion with constant acceleration.
• Suppose at time $t=0$ object is at origin of co-ordinate system and velocity components ${v_{0x}}$ and ${v_{0y}}$. From above components of acceleration are ${a_x} = 0$ and ${a_y} = - g$. From equation (11) and (12) in the previous section components of position and velocity are
$x = {v_{0x}}t$                                          (14a)
${v_x} = {v_{0x}}$                                           (14b)
and ${v_y} = {v_{0y}} - gt$                                          (15b)
$y = {v_{0y}} - \frac{1}{2}g{t^2}$                                          (15a)
• Figure below shows motion of an object projected with velocity ${v_0}$ at an angle ${\theta _0}$ .

• In terms of initial velocity ${v_0}$ and angle ${\theta _0}$ components of initial velocity are
${v_{0x}} = {v_0}cos{\theta _0}$                                          (16a)
${v_{0y}} = {v_0}sin{\theta _0}$                                           (16b)

• Using these relations in equation 14 and 15 we find
x=(v0cosθ0)t                                          (17a)
y=(v0sinθ0)t-(1/2)gt2                                           (17b)
vx=v0cosθ0                                           (17c)
vy=v0sinθ0-gt                                           (17d)

Above equations describe the position and velocity of projectile as shown in fig 5 at any time t.

(A)Equation of Path of projectile(Trajectory)
• From equation 17a
t=x/v0cosθ0
now putting this value of t in equation 17b,we find
y=(tanθ0 )x-[g/2(v0cosθ0)2]x2                                           (18)

In equation (18),quantities θ0,g and v0 are all constants and equation (18) can be compared with the equation
y=ax-bx2
where a and b are constants

• This equation y=ax-bx2 is the equation of the parabola.From this we conclude that path of the projectile is a parabola as shown in figure 5
B) Time of Maximum height
• At point of maximum height vy=0.Thus from equation (17d)
vy=v0sinθ0-gt
0=v0sinθ0-gt -
or tm=v0sinθ0/g                                          (19)
• Time of flight of projectile which is the total time during which the projectile is in flight can be obtained by putting y=0 because when projectile reaches ground ,verical distance travelled is zero.This from equation (17b)

tf=2(v0sinθ0)/g                                          (20)
or
tf=2tm
• Maximum height reached by the projectile can be calculated by substituting t=tm in equation 17b
y=Hm=(v0sinθ0)(v0sinθ0/g)-(g/2)(v0sinθ0/g)2
or
Hm=v02sin2θ0/2g                                          (21)
(C) Horizontal Range of Projectile
• Since acceleration g acting on the projectile is acting vertically ,so it has no component in horizontal direction.
• So, projectile moves in horizontal direction with a constant velocity v0cosθ0. So range R is
R=OA=velocity x time of flight

• Maximum range is obtained when sin2θ0=1 or θ0=450. Thus when θ0=450 maximum range achieved for a given initial velocity is (v0)2/g.