The Path of the Projectile

When a particle is projected horizontally from the top of a tower with a velocity v0, it follows a curved path due to the influence of gravity. This path is called the trajectory of the projectile.

The Velocity of the Projectile

The velocity of the projectile can be broken down into two components: the horizontal component (v_x) and the vertical component (v_y). Since the particle is projected horizontally, the initial vertical velocity is zero (v_y0 = 0). The horizontal velocity remains constant throughout the motion, while the vertical velocity changes due to the acceleration of gravity.

At any instant during the projectile's motion, the total velocity (v) can be calculated using the Pythagorean theorem:

v = √(v_x^2 + v_y^2)

The Radius of Curvature

The radius of curvature of the path of the particle at a given instant is a measure of how sharply the trajectory is curved at that point. It is the radius of the circle that best approximates the shape of the curve at that instant.

Proportional Relationship

Now, let's consider the relationship between the radius of curvature and the velocity of the particle. As the velocity of the particle increases, the radius of curvature decreases, indicating a sharper curve. Conversely, as the velocity decreases, the radius of curvature increases, indicating a gentler curve.

Applying the Options

a) v^2: If the radius of curvature is directly proportional to v^2, it means that as the velocity increases, the radius of curvature increases. However, we know that the radius of curvature actually decreases with increasing velocity. Therefore, option 'a' is incorrect.

b) 1/v: If the radius of curvature is directly proportional to 1/v, it means that as the velocity increases, the radius of curvature decreases. This is in line with what we just discussed. Therefore, option 'b' is incorrect.

c) v: If the radius of curvature is directly proportional to v, it means that as the velocity increases, the radius of curvature increases. However, we know that the radius of curvature actually decreases with increasing velocity. Therefore, option 'c' is incorrect.

d) v^3: If the radius of curvature is directly proportional to v^3, it means that as the velocity increases, the radius of curvature decreases. This matches our understanding of the relationship between velocity and the radius of curvature. Therefore, option 'd' is correct.

Conclusion

The correct option is 'd' because the radius of curvature of the path of the projectile at any instant is directly proportional to v^3. As the velocity of the particle increases, the radius of curvature decreases, indicating a sharper curve.