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.

Figure 5: Vertical component of velocity during motion

Projectile motion

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 y v 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 30 0 with the horizontal. Here, vertical component of velocity is 60 sin 30 0 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               20
2        20            10               10
3        30             0                0
4        40           -10               10
5        50           -20               20
6        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.

Figure 6: The velocity – time plot for constant acceleration in vertical direction

Velocity – time plot

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

The time to reach maximum height in this case is :

⇒ t = u y g = 30 10 = 3 s ⇒ t = u y g = 30 10 = 3 s

The data in the table confirms this. Further, we know that vertical motion is independent of horizontal motion and time of flight for vertical motion is equal for upward and downward journey. This means that total time of flight is 2t i.e. 2x3 = 6 seconds. We must, however, be careful to emphasize that this result holds if the point of projection and point of return to the surface are on same horizontal level.

Figure 7: Initial and final velocities are equal in magnitude but opposite in direction.

Projectile motion

There is yet another interesting feature that can be drawn from the data set. The magnitude of vertical velocity of the projectile (30 m/s) at the time it hits the surface on return is equal to that at the time of the start of the motion. In terms of velocity, the final vertical velocity at the time of return is inverted initial velocity.

Displacement

The displacement in vertical direction is given by :

Figure 8: Vertical displacement at a given time

Projectile motion

This equation gives vertical position or displacement at a given time. It is important to realize that we have simplified the equation Δ y = y 2 - y 1 = u y t - 1 2 g t 2 Δ y = y 2 - y 1 = u y t - 1 2 g t 2 by selecting origin to coincide by the point of projection so that,

Δ y = y 2 - y 1 = y ( say ) Δ y = y 2 - y 1 = y ( say )

Thus, “y” with this simplification represents position or displacement.

The equation for position or displacement is a quadratic equation in time “t”. It means that solution of this equation yields two values of time for every value of vertical position or displacement. This interpretation is in fine agreement with the motion as projectile retraces all vertical displacement as shown in the figure.

Figure 9: All vertical displacement is achieved twice by the projectile except the point of maximum height.

Projectile motion

In case, initial and final points of the journey are on the same horizontal level, then the net displacement in vertical direction is zero i.e. y = 0. This condition allows us to determine the total time of flight “T” as :

y = u y T - 1 2 g T 2 ⇒ 0 = u y T - 1 2 g T 2 ⇒ T ( u y - 1 2 g T ) = 0 ⇒ T = 0 or T = 2 u y g y = u y T - 1 2 g T 2 ⇒ 0 = u y T - 1 2 g T 2 ⇒ T ( u y - 1 2 g T ) = 0 ⇒ T = 0 or T = 2 u y g

T = 0 corresponds to the time of projection. Hence neglecting the first value, the time of flight is :

We see that total time of flight is half of the time projectile takes to reach the maximum height. It means that projectile takes equal times in "up" and "down" motion. In other words, time of ascent equals time of descent.