Two balls are projected from the same point in
directions inclined at 60 and 30 to the
horizontal. If they attain the same maximum
height, the ratio of their velocities of projection
is :
A) 1 : 1 B) 1 : 2 C) 1 :
p
3 D)
p
3 : 1

To find the ratio of the velocities of projection of the two balls, we can use the relationship between the maximum height attained by a projectile and its initial velocity. The maximum height \( H \) of a projectile is given by the formula:

Understanding Maximum Height

The maximum height \( H \) reached by a projectile can be expressed as:

H = (u^2 * sin^2(θ)) / (2g)

where:

• u is the initial velocity of projection.
• θ is the angle of projection.
• g is the acceleration due to gravity (approximately 9.81 m/s²).

Applying the Formula

In this case, we have two angles of projection: 60° and 30°. Let’s denote the initial velocities of the two balls as \( u_1 \) and \( u_2 \). According to the problem, both balls reach the same maximum height. Therefore, we can set the maximum heights equal to one another:

H_1 = H_2

Substituting the formula for maximum height, we have:

(u_1^2 * sin^2(60°)) / (2g) = (u_2^2 * sin^2(30°)) / (2g)

Calculating the Sine Values

Now, let's calculate the sine values:

• sin(60°) = √3/2
• sin(30°) = 1/2

Substituting Values

Substituting these values into the height equation gives:

(u_1^2 * (√3/2)^2) = (u_2^2 * (1/2)^2

Which simplifies to:

(u_1^2 * 3/4) = (u_2^2 * 1/4)

Solving for the Ratio

Now we can eliminate the fractions by multiplying both sides by 4:

3u_1^2 = u_2^2

Taking the square root of both sides provides:

u_2 = √3 * u_1

Finding the Ratio

To find the ratio of the velocities of projection, we express it as:

u_1 : u_2 = u_1 : (√3 * u_1) = 1 : √3

This can also be expressed as:

u_1 : u_2 = 1 : p3

Conclusion

Therefore, the correct answer to the ratio of their velocities of projection is option C) 1 : √3.