A point source S is moving with speed 10 m/s in x-y plane at angle of 37 degree. The radius of the concave mirror is 4m. Determine the velocity vector if object distance from mirror is 6m and object is at height of 1m

To determine the velocity vector of the image formed by a point source moving towards a concave mirror, we need to consider the principles of optics, particularly the mirror formula and the concept of magnification. Let's break this down step by step.

Understanding the Setup

We have a point source S moving at a speed of 10 m/s at an angle of 37 degrees in the x-y plane. The radius of the concave mirror is 4 m, which gives us a focal length (f) of 2 m (since the focal length is half the radius for a spherical mirror). The object distance (u) from the mirror is given as 6 m, and the height of the object is 1 m.

Calculating the Focal Length

The focal length (f) of a concave mirror is calculated as:

• f = R/2

Here, R is the radius of the mirror. So, substituting the values:

• f = 4 m / 2 = 2 m

Using the Mirror Formula

The mirror formula relates the object distance (u), image distance (v), and focal length (f) as follows:

• 1/f = 1/v + 1/u

Substituting the known values into the formula:

• 1/2 = 1/v + 1/6

To solve for v, we first find a common denominator:

• 1/2 = 3/6 + 1/v

Rearranging gives us:

• 1/v = 1/2 - 1/6 = 3/6 - 1/6 = 2/6 = 1/3

Thus, we find:

• v = 3 m

Determining the Velocity Vector

Now, we need to find the velocity vector of the image. The object is moving with a velocity of 10 m/s at an angle of 37 degrees. We can break this velocity into its x and y components:

• Vx = 10 * cos(37°)
• Vy = 10 * sin(37°)

Calculating these components:

• Vx ≈ 10 * 0.798 = 7.98 m/s
• Vy ≈ 10 * 0.601 = 6.01 m/s

Applying the Magnification Factor

The magnification (m) of the mirror is given by:

• m = -v/u = -3/6 = -0.5

This means the image is inverted and half the height of the object. The height of the image (h') can be calculated as:

• h' = m * h = -0.5 * 1 m = -0.5 m

Calculating the Image Velocity

The velocity of the image (Vi) can be found using the relationship:

• Vi = m * Vo

Where Vo is the object velocity vector. Thus:

• Vi_x = -0.5 * Vx = -0.5 * 7.98 ≈ -3.99 m/s
• Vi_y = -0.5 * Vy = -0.5 * 6.01 ≈ -3.00 m/s

Final Velocity Vector

Combining these components, the velocity vector of the image is:

• Vi = (-3.99, -3.00) m/s

This indicates that the image is moving in the opposite direction to the object, consistent with the properties of concave mirrors. The negative signs indicate that the image is inverted and moving downwards in the y-direction.

In summary, the velocity vector of the image formed by the concave mirror, given the conditions of the problem, is approximately (-3.99, -3.00) m/s.