Projectile motion  (Page 3/6)

Velocity

The velocity in the vertical direction is given by :

An inspection of equation - 2 reveals that this equation can be used to determine velocity in vertical direction at a given time “t” or to determine time of flight “t”, if final vertical velocity is given. This assumes importance as we shall see that final vertical velocity at the maximum height becomes zero.

The equation for velocity further reveals that the magnitude of velocity is reduced by an amount “gt” after a time interval of “t” during upward motion. The projectile is decelerated in this part of motion (velocity and acceleration are in opposite direction). The reduction in the magnitude of velocity with time means that it becomes zero corresponding to a particular value of “t”. The vertical elevation corresponding to the position, when projectile stops, is maximum height that projectile attains. For this situation ( $v_{\mathrm{y}}$ = 0), the time of flight “t” is obtained as :

Immediately thereafter, projectile is accelerated in vertically downward direction with increasing speed. In order to appreciate variation of speed and velocity during projectile motion, we calculate the values of a projectile for successive seconds, which is projected with an initial velocity of 60 m/s making an angle of ${30}^0$ with the horizontal. Here, vertical component of velocity is 60 sin ${30}^0$ " i.e. 30 m/s.

-------------------------------------------------------- Time gt Velocity Magnitude of velocity(s) (m/s) (m/s) (m/s) --------------------------------------------------------0 0 30 30 1 10 20 202 20 10 10 3 30 0 04 40 -10 10 5 50 -20 206 60 -30 30 --------------------------------------------------------

Above table substantiate the observations made earlier. The magnitude of vertical velocity of the projectile first decreases during upward flight; becomes zero at maximum height; and, thereafter, picks up at the same rate during downward flight.

It is also seen from the data that each of the magnitude of vertical velocity during upward motion is regained during downward motion. In terms of velocity, for every vertical velocity there is a corresponding vertical velocity of equal magnitude, but opposite in direction.

The velocity – time plot of the motion is a straight line with negative slope. The negative slope here indicates that acceleration i.e acceleration due to gravity is directed in the opposite direction to that of positive y- direction.

From the plot, we see that velocity is positive and acceleration is negative for upward journey, indicating deceleration i.e. decrease in speed.