To analyze the problem, consider two objects:
First Object: Maximizing Range
To achieve maximum range, the object must be projected at a
45∘ angle.
Using the formula for maximum height
H1 when launched at an angle
θ1=45∘ :
H1=‌=‌=‌Second Object: Maximizing Height
To achieve maximum height, the object should be projected vertically at a
90∘ angle.
The formula for the maximum height
H2 with a launch angle
θ2=90∘ :
H2=‌=‌Equating Heights
Since both objects reach the same maximum height:
H1=H2From equations (i) and (ii):
‌=‌Simplifying, we find:
u12=2u22Taking the square root:
u1=√2u2Thus, the ratio of their initial velocities is:
‌=‌Therefore, the ratio of initial velocities is
√2:1.
