Velocity of Image in Spherical Mirror: Formula & Calculation Guide
Velocity Of Object And Image is a key topic in ray optics that explains how the motion of an object changes the motion of its image in mirrors and lenses. For example, if you walk towards a mirror, your image appears to move towards you at a certain speed. Understanding these velocity relationships is crucial for solving JEE Main optics numericals efficiently.
This concept combines image formation, sign conventions, and differentiation. Calculating image speed in spherical mirrors, lenses, and during refraction requires careful use of formulas and direction analysis. Mistakes usually happen if you ignore signs or misunderstand real versus virtual image motion.
Foundations of Velocity Of Object And Image in Optics
When an object moves relative to a mirror or lens, its image also moves. The direction and speed depend on the type of optical element and the chosen sign convention. Always use the principal axis as reference and measure distances from the mirror or lens pole.
For spherical mirrors, the relationship is derived using the mirror formula. Lenses follow similar principles but with different sign rules. The relative motion between object and image (sometimes called 'relative velocity in optics') matters in problems like moving car mirrors or running water droplets near a lens.
- Object speed is usually denoted as vo.
- Image speed is vi.
- Distances use u (object), v (image), and f (focal length).
- Pay attention to direction (towards or away from mirror/lens).
- Apply sign conventions, especially for concave and convex elements.
You can strengthen your basics from related topics such as sign convention in optics and lenses before working with velocities.
Key Formulas for Velocity Of Object And Image
The essential formula to relate an object's velocity and the image's velocity in a spherical mirror (using standard sign conventions) is:
| Optical System | Velocity Relation | Notes (Sign & Direction) |
|---|---|---|
| Plane Mirror | vi = -vo | Image moves at same speed, opposite direction |
| Spherical Mirror | vi = (m2)vo | m = magnification = v/u; mind sign |
| Lens (Thin) | vi = (v2/u2)vo | Consistent sign convention needed |
| Refraction at Spherical Surface | Use formula derivation method | Dependent on medium and surface curvature |
vo is the object velocity, vi is the image velocity, and m is the magnification. The negative sign denotes opposite direction (as in plane mirrors).
For more on magnification and image properties, see magnification concepts and image type differences.
Stepwise Derivation: Velocity Of Object And Image in Spherical Mirrors
Let's derive the velocity relationship for a spherical mirror. The mirror formula is:
1/v + 1/u = 1/f
- Differentiate both sides with respect to time to relate velocities.
- d/dt (1/v + 1/u) = 0, since f is constant for the mirror.
- That gives: -(1/v2)vi - (1/u2)vo = 0
- Rearrange for image velocity: vi = (v2/u2)vo
- Since m = v/u, then vi = m2 vo
Sign convention is crucial: for a mirror, negative u means object on the left (usual setup). For more on the sign rules, use sign convention for mirrors and lenses.
Understanding Direction and Diagrams: Velocity Of Object And Image
The motion direction of images can be visualized using ray diagrams. For instance, if an object moves towards a concave mirror, its image moves towards or away depending on its position relative to the center of curvature.
- If the object approaches the mirror, the image rapidly increases in speed near the focus.
- In a convex lens, as the object crosses the focal point, the image flips direction.
- Speed of image can be much greater than object when near focal points.
- Direction of image velocity is found by careful sign analysis.
- In one-dimensional motion, just consider axis movement; in 2D, resolve components.
Practice drawing ray diagrams and arrows for objects and images to strengthen this skill.
Numerical Example: Applying Velocity Of Object And Image Formulas
A point object moves towards a concave mirror of focal length f = 15.0 cm at vo = 4.0 cm/s. Find the image speed when the object is u = -30.0 cm from the mirror.
- Mirror formula: 1/v + 1/u = 1/f
- Insert values: 1/v + 1/(-30) = 1/(-15)
- Solve for v: 1/v = 1/(-15) - 1/(-30) = (-2 + 1)/30 = -1/30
- v = -30 cm (so magnification m = v/u = -30/-30 = +1)
- vi = m2 vo = (1)2 × 4.0 = 4.0 cm/s
Thus, the image moves at 4.0 cm/s towards the mirror in this case. Try other positions, or attempt questions from this optics practice paper.
Common Pitfalls: Velocity Of Object And Image
- Ignoring the sign convention when substituting values.
- Assuming image direction is always the same as object motion.
- Forgetting that speed can be much larger or smaller than object.
- Mistaking the use of v versus u in formulas.
- Not analyzing 2D motion when velocity vectors are angled.
- Confusing real and virtual image behavior near focal points.
To avoid mistakes, check your signs, directions, and magnitudes at each step. For extra clarity, refer to image characteristics.
Summary Table: Velocity Of Object And Image Cases
| Case | Formula | Key Direction Rule |
|---|---|---|
| Plane Mirror | vi = -vo | Always opposite to object |
| Spherical Mirror | vi = (m2)vo | Use sign of m to check |
| Thin Lens | vi = (v2/u2)vo | Direction needs axis check |
Use this table for fast revision before your JEE Main exam. Revisiting the mirror formula and magnification strengthens your recall and application skills.
For more related concepts and mock tests, explore optics revision notes, motion in two dimensions, and relative motion in the context of ray optics. Practicing a variety of problems ensures you are ready for any velocity of image question in the JEE Main.
Content crafted and reviewed by experienced Vedantu JEE Physics educators—supporting you in mastering concepts like velocity of object and image with clarity and exam confidence.
FAQs on Velocity of Object and Image in Mirrors and Lenses
1. What is the relation between velocity of object and image in case of a spherical mirror?
The velocity of image in a spherical mirror is related to object velocity by the mirror's magnification squared. This relationship is given by:
- vi = (m2) × vo, where vi is image velocity, vo is object velocity, and m is magnification.
- For concave and convex mirrors, pay attention to the sign convention for direction.
- Use the mirror and magnification formulas for numerical calculation in exam questions.
- This concept is essential for JEE, NEET, and CBSE optics syllabus.
2. What is the formula to calculate velocity of image in spherical mirrors?
The formula for image velocity in spherical mirrors is:
- vi = (v2 / u2) × vo
- Here,
• vi = velocity of image,
• vo = velocity of object,
• u = object distance,
• v = image distance.
- Alternatively, vi = (magnification)2 × vo can be used.
- Always use the correct sign convention for directions in numericals.
3. How is velocity of image related to object velocity in a plane mirror?
In a plane mirror, the velocity of the image equals the velocity of the object but in the opposite direction along the normal.
- If the object moves towards the mirror with speed v, the image moves towards the mirror with speed v relative to the mirror.
- Relative to the object, the image recedes at 2v.
- Direction and sign conventions are important for solving MCQs and numericals in JEE mains and boards.
4. What is the relationship between object speed and image speed in lens?
The image speed in a lens depends on the lens formula and object movement direction.
- Use the lens formula: 1/v - 1/u = 1/f, where u = object distance, v = image distance, and f = focal length.
- Differentiate both sides with respect to time to relate object velocity (du/dt) and image velocity (dv/dt).
- The final relation: vi = (v2/u2) × vo for thin lenses.
- Sign convention and direction are vital for correct answers.
5. How do you apply sign conventions for velocity in lenses and mirrors?
Follow the New Cartesian Sign Convention for optics velocity problems:
- Distances measured towards the mirror/lens (left of the pole/optical center) are negative.
- Distances away from the mirror/lens (right) are positive.
- Velocities in the same direction as the incident light are usually positive; opposite is negative.
- Always check the direction of motion and mark signs accordingly for avoiding calculation errors in exams.
6. What happens to image velocity if the object moves towards the mirror?
If the object approaches the mirror, its image also moves towards or away depending on mirror type:
- Plane mirror: Image moves towards mirror at the same speed.
- Concave or convex mirror: Image velocity is determined by the mirror formula and magnification (can be faster or slower).
- Relative speed between object and image can be up to twice the object speed in special cases.
7. Can the direction of image velocity be opposite to the object's motion?
Yes, the image velocity direction can be opposite to the object’s motion depending on mirror or lens type.
- For a plane mirror, the image always follows the object’s motion, maintaining the same speed but in a symmetrical position.
- In spherical mirrors and lenses, due to image formation rules and sign conventions, image motion can be in the opposite direction, especially when the object crosses the focal point or center of curvature.
8. Are velocity formulas different for real versus virtual images?
The mathematical form of the velocity formula remains the same for real and virtual images, but the sign convention changes.
- Apply sign convention rules for image type (real: generally formed on the same side; virtual: opposite side), as this affects whether velocities are positive or negative.
- This distinction is vital for correct answers in JEE, NEET, and board exams.
9. What errors can occur if I ignore sign convention in numericals?
Ignoring sign conventions can lead to incorrect calculation of velocity values and wrong direction prediction.
- Possible errors include:
• Getting the wrong magnitude or sign for the image velocity
• Mistaking direction (towards or away from mirror/lens)
• Losing marks due to unit or sign mistakes in exams
- Always apply the correct sign convention when solving optics motion questions.
10. What is the velocity of image for a moving object in a concave mirror?
The velocity of the image in a concave mirror varies based on object location and motion direction:
- Use the formula vi = (v2/u2) × vo, or vi = (magnification)2 × vo.
- Substitute sign according to whether the object is in front of the mirror and its movement direction.
- Apply the concave mirror magnification and sign convention for full credit in board and competitive exams.
