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The span of projectile motion in the vertical plane is determined by two factors, namely the speed of projection and angle of projection with respect to horizontal. These two factors together determine (i) how long does the projectile remain in air (time of flight, T) (ii) how far does the projectile go in the horizontal direction (range of projectile, R) and (iii) how high does the projectile reach (maximum height, H).
Further, the trajectory of the projectile is symmetric about a vertical line passing through the point of maximum height if point of projection and point of return fall on the same horizontal surface.
We have already determined the time of flight, which is given by :
This equation was derived in the earlier module Projectile motion with the assumption that both point of projection and point of return of the projectile lie on same horizontal level. It may be also be recalled that the equation of motion in vertical direction was evaluated for the condition that net displacement during the entire motion is zero. Hence, if the points are not on the same level, then above equation will not be valid and must be determined by equation of motion for the individual case with appropriate values.
From the above equation, we see that time of flight depends on initial speed and the angle of projection (θ). We must realize here that the range of θ is 0° ≤ θ ≤ 90°. For this range, sinθ is an increasing function. As such, we can say that a projection closer to vertical direction stays longer in the air for a given initial velocity. As a matter of fact, a vertical projectile for which θ = 90° and sinθ = 1, stays in the air for the maximum period.
If the points of projection and return are on same level and air resistance is neglected, which of the following quantities will enable determination of the total time of flight (T) :
(a) horizontal component of projection velocity
(b) projection speed and angle of projection
(c) vertical component of projection velocity
(d) speed at the highest point
The total time of flight is given by :
We can see that the total time of flight can be determined if vertical component of the velocity ( ) is given. Hence option (c) is correct. The vertical component of the velocity ( ) , in turn, is determined by the projection speed (u) and angle of projection ( ). Hence option (b) is correct.
Now speed at the highest point is equal to the horizontal component of projection velocity (u cos ). We can not, however, determine vertical component (u sin ) from this value, unless either "u" or " " is also given.
Hence, options (b) and (c) are correct.
Note: We have noticed that time of flight is derived considering vertical motion. Horizontal part of the motion is not considered. Thus, the time of flight (t) at any point during the projectile motion is dependent on vertical component of velocity or vertical part of the motion and is independent of horizontal part of the motion.
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