NCERT 11th Class : Motion in a Plane : Projectile Motion

Projectile : An object that is in flight after being thrown or projected is called a projectile. A stone thrown into a pond is an example of a projectile. The motion of a projectile is the result of two separate, simultaneously occurring components of motions (vertical and horizontal). One component is along a horizontal direction without any acceleration and the other along the vertical direction with constant acceleration due to the force of gravity. It was Galileo who first stated this in-dependency of the horizontal and the vertical components of projectile motion in his Dialogue on the great world systems (1632). 

Velocity components of a projectile

 

After the object has been projected, the acceleration acting on it is that due to gravity which is directed vertically downward: a j = −g 

Or, a _x = 0, a _y = – g

The components of initial velocity v_o are :

v_ox= v_ocos θ_o

        v_oy= v_osinθ_o.

If we take the initial position to be the origin of the reference frame.

x₀ = 0, y₀ = 0 

Then x = v_ocos θ_o t

y = v_osin θ_o t - 1/2gt²

i.e. v_x = v_ocos θ_o

      _Vy = v_osin θ_o - gt

Equation of path of a projectile

Time of maximum height : Let t_m be the time taken for the projectile to reach the maximum height. 

Since at this point v_y = 0, we have v_y = v_osin θ_o - gt_m

0 = v_osin θ_o - gt_m

_{tm =} v_osinθ_o /g

Time of flight of a projectile : The total time T_f during which the projectile is in flight can be obtained by putting y = 0 in y = v_osin θ_o t - 1/2gt²

0 = v_osin θ_oT_f - 1/2gT_f ²

T_f = 2v_osin θ_o/g = 2t_m

Maximum height of a projectile : The maximum height hm reached by the projectile can be calculated by substituting t = t_m in the equation for y component.

 

Horizontal range of a projectile : The horizontal distance travelled by a projectile from its initial position (x = y = 0) to the position where it passes y = 0 during its fall is called the horizontal range, R. It is the distance travelled during the time of flight T_f.