a particle is dropped along the axis from a height f/2 on a concave mirror of focal lenght f. The acceleration due to gravity is g. Find maximum speed of image (pls show with steps)

To find the maximum speed of the image formed by a particle dropped from a height of f/2 above a concave mirror with a focal length of f, we need to consider both the motion of the particle and the properties of the mirror. Let's break this down step by step.

Understanding the Motion of the Particle

When the particle is dropped, it falls under the influence of gravity. The equation of motion for an object in free fall can be expressed as:

• h = (1/2)gt²

Here, h is the height from which the particle is dropped, g is the acceleration due to gravity, and t is the time taken to fall. Since the particle is dropped from a height of f/2, we can set h = f/2:

• f/2 = (1/2)gt²

Rearranging this gives:

• t² = f/g
• t = √(f/g)

Calculating the Speed of the Particle

The speed of the particle just before it hits the mirror can be found using the formula:

• v = gt

Substituting the expression for t we derived earlier:

• v = g(√(f/g))
• v = √(gf)

Image Formation by the Concave Mirror

Now, let's consider how the concave mirror affects the image of the falling particle. The mirror's focal length is f, and the object (the particle) is falling towards the mirror. As the particle approaches the mirror, it acts as a point source of light.

For a concave mirror, the relationship between the object distance (u), image distance (v), and focal length (f) is given by the mirror formula:

• 1/f = 1/v + 1/u

As the particle falls, its distance from the mirror (u) decreases. When the particle is at the mirror's surface, u = -f/2 (the negative sign indicates that the object is in front of the mirror). Substituting this into the mirror formula:

• 1/f = 1/v + 1/(-f/2)

Solving for v gives:

• 1/v = 1/f + 2/f = 3/f
• v = f/3

Finding the Maximum Speed of the Image

The speed of the image (v_image) can be related to the speed of the object (v) using the magnification formula for mirrors:

• m = -v_image/v

Since the magnification (m) is also equal to the ratio of the image distance to the object distance:

• m = v/u = v/(−f/2) = −2v/f

Setting the two expressions for magnification equal gives:

• -v_image/v = -2v/f

From this, we can solve for v_image:

• v_image = 2v(f/v) = 2f

Now substituting v = √(gf) into this equation:

• v_image = 2√(gf)

Final Result

The maximum speed of the image formed by the particle just before it hits the concave mirror is:

• v_image = 2√(gf)

This result shows how the speed of the image is directly influenced by both the gravitational acceleration and the focal length of the mirror. The faster the particle falls, the faster the image moves, illustrating the dynamic relationship between object motion and image formation in optics.