Time of Flight and Horizontal-range Physics Notes

Time of Flight and Horizontal-range Physics Notes

Time of Flight:
It is the total time taken by the projectile when it is projected from a point and reaches the same horizontal plane or the time for which the projectile remains in the air above the horizontal plane.

It is denoted by T.
As the motion from the point O to A and then from point A to B are symmetrical, the time of ascent (for the journey from point O to A) and the time of descent (for the return journey from A to B) will be each equal to T/2.

At the highest point A, the vertical component of velocity of the object becomes zero. Taking vertically upward motion of the object from O to A, we have

The time of flight is independent of the horizontal component of velocity. The faster a projectile is thrown up, the longer it will stay is the air.

Maximum height of a projectile:
It is the maximum vertical height attained by the object above the point of projection during its flight. It is denoted by H.

Taking the vertical upward movement of the object 122 (B)from O to A, we have:

1. The maximum height is independent of the horizontal component of velocity. The faster a projectile is thrown upwards, the higher it will go in an upward direction, i.e., the longer it will resist the downward pull of gravity.

2. Both time of flight and maximum height depends upon the vertical component of velocity, thus the relation between them can be expressed as
\(\frac{H}{T^{2}}=\frac{g}{8}\)

Horizontal-range:
It is the horizontal distance covered by the object between its point of projection and the point of hitting the ground. It is denoted by R.

Obviously, the horizontal range R is the horizontal distance covered by the projectile with the’ uniform velocity u cos θ in a time equal to the time of flight.

The angle of projection for maximum range:
The value of the horizontal range depends upon the angle of projection O. Therefore, horizontal range R will be maximum if
sin 2θ = maximum = 1 = sin90°
or 2θ = 900
or θ = 45°
∴ Maximum horizontal range, Rmax = \(\frac{u^{2}}{g}\) …..(7)

1. The horizontal range depends upon both the horizontal and vertical components of velocity.
2. For a specified speed of projection, the range will maximum at an angle of projection equal to 45°
3. Projectile moving at equal speed, the range will be equal when both projectiles have a complementary angle of projection. it means θ or 90-θ

Physics Notes