Oblique Projectile.

In projectile motion, horizontal component of velocity (u cos?), acceleration (g) and mechanical energy remains constant while, speed, velocity, vertical component of velocity (u sin ?), momentum, kinetic energy and potential energy all varies with time. Velocity, and Kinetic Energy are maximum at the point of projection while minimum (but not zero) at highest point.

Time of flight : The total time taken by the projectile to move from starting point to final point is called time of flight.

For vertical upward motion(Along Y axis)

v_y=u_y + a_yt  

 0 = u sin? – gt  

  t = (u sin ?  /g)

Now as time taken to go up(time of accent) is equal to the time taken to come down(time of descent) so

Time of flight         T=2 t = (2u sin ?/g)

 Time of flight can also be expressed as : T= 2 u_y/g  (where u_y is the vertical component of initial velocity).

 

In general, T = 2v_?/a_?

Where v_? and a_? are the respective velocity and acceleration perpendicular to the surface.

Note: that if a projectile is thrown up an an inclined plane as shown in the fig.

Here      v_? = v_o sinθ

And        a_? = g cos α

Thus,     T = (2v_?/a_?)=(2v_o sinθ/g cos α)

 

Instantaneous velocity v : Let v_i be the instantaneous velocity of projectile at time t direction of this velocity is along the tangent to the trajectory at point P.