Question

There are two initial velocities for which a given projectile has the same range? How many initial velocities are there to have the same maximum height? The same time of flight?

Answer 1

The number of initial velocities with the same range of projectiles:

1. The formula for the range of a projectile is: $R=\frac{u^2\sin 2\theta }{g}$
2. Two projectiles can have the same range if their initial speed is the same but the angles of projection are $\theta$ and $(90-\theta )$.
3. Mathematically,
   $$\begin{gathered} R_1=\frac{u^2\sin 2\theta }{g} \\ {R}_2=\frac{u^2\sin 2(90-\theta )}{g} \\ =\frac{u^2\sin (180-2\theta )}{g} \\ =\frac{u^2\sin 2\theta }{g} \\ =R_1 \end{gathered}$$
4. Even though the magnitude is the same, the direction of the angle is different which means that the velocity of both the projectiles is different.

The number of initial velocities with the same maximum height of projectiles:

1. The formula for the maximum height of a projectile is: $H=\frac{u^2{\sin }^2\theta }{2g}$
2. The maximum height of a projectile depends upon the initial velocity and the angle of projection.
3. There can be multiple combinations of the magnitude of initial velocity and the angle of projections for the same value.
4. Thus, we can have multiple numbers of different initial velocities for the same maximum height.
5. There is no fixed number or well-defined rule for it.

The number of initial velocities with the same time of flight of projectiles:

1. The formula for the time of flight of a projectile is: $T=\frac{2u\sin \theta }{g}$
2. The time of flight of a projectile depends upon the initial velocity, and the angle of projection.
3. There can be multiple combinations of the magnitude of initial velocity and the angle of projections for the same value.
4. Thus, we can have multiple cases of different initial velocities for the same time of flight.
5. There is no fixed number or well-defined rule for it.

Hence, there are multiple initial velocities for which a projectile can have the same maximum height and time of flight.