How to Calculate the Velocity of an Image Formed by a Mirror or Lens
Velocity of Object and Image describes how the movement of an object in front of a mirror or lens determines the motion of its image. This idea is essential in ray optics for JEE Main as it combines formulas, sign conventions, and the physics of image formation.
Suppose a car moves towards a plane mirror on a straight road. Its image also moves, but in which direction and at what speed? Understanding such cases helps solve optics numericals fast and accurately.
Key Formulas for Velocity of Object and Image
JEE problems often require a direct link between object and image velocities. Here are the core relationships.
- Plane mirror: vi = -vo (opposite direction, equal speed)
- Spherical mirror: vi = (m2) vo (m = v/u, sign matters)
- Thin lens: vi = (v2/u2) vo
Where vi is image velocity, vo is object velocity, v is image distance, u is object distance. Always apply the correct sign convention as per the New Cartesian system.
For a better sense of the concept, check how speed and direction relate for objects and images using a clear example with plane mirrors.
Derivation for Spherical Mirror Velocity Formula
Step 1: Start with the mirror formula
1/v + 1/u = 1/f
Step 2: Differentiate with respect to time (t)
d/dt(1/v) + d/dt(1/u) = 0
(-1/v2)(dv/dt) + (-1/u2)(du/dt) = 0
Step 3: Rearranging for image velocity
(1/v2)(vi) = -(1/u2)(vo)
Step 4: Expressing vi in terms of vo
vi = (v2/u2) vo
Final expression: vi = (v2/u2) vo
This derivation shows that the image velocity can be greater or opposite in direction to object velocity, depending on the sign and values of u and v. This is crucial for concave and convex mirrors, especially near the focal point or center of curvature.
Physical Meaning and Direction Analysis
Imagine a ball moving towards a concave mirror. If it is beyond the center of curvature, its image moves towards the mirror, but not always at the same rate. As the object nears the focus, even a small motion creates a rapid image shift.
For convex lenses, when the object crosses the focal point, the image jumps sides, leading to a sign reversal in image velocity. This behaviour is often tested in JEE as a test of sign convention mastery.
- Image and object may move in opposite directions.
- Magnitude of image velocity can exceed that of the object.
- Careful vector and sign tracking is needed in two-dimensional cases.
Applying the correct sign convention for lenses and mirrors prevents common mistakes.
Velocity of Object and Image: Exam-Ready Examples
Let’s apply the formulas to typical JEE level scenarios. Consider these solved cases:
| Case | Formula | Direction Rule |
|---|---|---|
| Plane mirror | vi = -vo | Opposite to object motion |
| Concave mirror | vi = (v2/u2) vo | Sign shows image direction |
| Convex lens | vi = (v2/u2) vo | Test direction for axis sign |
For actual JEE questions, always substitute examiner-provided u, v, and vo, using negative numbers for objects in front of mirrors, and check direction using ray diagrams.
When you see terms like “velocity of image” or “object speed in lens” in a problem, break it down by formula first, then confirm direction using the sign convention. Many JEE numericals differentiate students by these details.
Common Mistakes in Using Velocity Formulas
JEE toppers avoid these pitfalls when tackling velocity relationships in optics:
- Forgetting to switch signs for motion direction
- Mixing up v (image) and u (object) in equations
- Ignoring vector angles in 2D motion setups
- Mistaking relative speed for actual speed
- Assuming all images move in the same way as objects
When confused, sketching ray diagrams and writing out each sign explicitly helps avoid errors.
For a detailed walk-through of 2D motion, explore motion in two dimensions here.
Summary Table: Velocity of Object and Image
Here’s a practical reference for the velocity relations across major optical elements relevant to JEE Main syllabus:
| Optical System | Core Formula | Direction Hint |
|---|---|---|
| Plane Mirror | vi = -vo | Always opposite |
| Spherical Mirror | vi = (m2) vo | Follow sign of magnification |
| Thin Lens | vi = (v2/u2) vo | Sign determines direction |
This topic represents a classic blending of kinematics and optics, and Vedantu’s physics experts ensure that clarity in these basics builds exam confidence.
Mastering the velocity of object and image not only scores marks in direct questions, but also sharpens your handling of ray diagrams and sign conventions throughout optics problems in JEE Main.
FAQs on Understanding Object and Image Velocity in Optics
1. What is the velocity of an image when the object moves towards a concave mirror?
The velocity of the image created by a concave mirror depends on the object’s velocity and its distance from the mirror’s focal point. When the object moves towards the mirror, the image velocity can be calculated using the mirror formula and differentiation:
Key Points:
- If the object moves towards the mirror with velocity v_o, the velocity of the image v_i is determined by the mirror equation: 1/f = 1/v + 1/u.
- Using differentiation, v_i = (v_o * v^2) / u^2, where u is object distance and v is image distance.
- For concave mirrors, image velocity is usually greater in magnitude and opposite in direction to the object’s velocity.
2. How do you find the velocity of the image formed by a lens or mirror?
The velocity of the image is found by differentiating the lens or mirror formula with respect to time.
Steps:
- Start with the basic mirror or lens formula: 1/f = 1/v + 1/u.
- Differentiating both sides with respect to time gives: 0 = -1/v2 (dv/dt) - 1/u2 (du/dt).
- Rearrange to find dv/dt = - (v2/u2) (du/dt).
- Here, (du/dt) is the velocity of the object; (dv/dt) is the velocity of the image.
3. What happens to the image velocity if the object is moving away from a convex mirror?
When an object moves away from a convex mirror, the image also moves away but at a different (smaller) velocity.
- The image formed in a convex mirror is always virtual, erect, and diminished.
- The magnitude of the image’s velocity is less than that of the object.
- If object velocity = v_o, image velocity = (v^2/u^2) * v_o (using mirror equation with proper sign conventions).
4. What is the significance of sign convention when calculating image velocity?
Sign convention determines the direction and nature of velocities in mirror and lens formulas, ensuring accurate results.
- Distances measured in the direction of incident light are positive; opposite directions are negative (Cartesian sign convention).
- For mirrors: object distances (u) are negative if placed in front; image distances can be positive or negative depending on the image location.
- Applying correct signs helps avoid calculation errors and ensures physically meaningful answers.
5. How do you solve numerical problems involving velocity of image formation?
To solve numerical problems about image velocities, use the differentiated form of the mirror/lens equation and insert all values with correct signs.
- Write the original formula (1/f = 1/v + 1/u).
- Differentiating w.r.t time: 0 = -1/v2(dv/dt) - 1/u2(du/dt).
- Insert given object velocity, object and image distances with sign conventions.
- Calculate the required image velocity (dv/dt).
6. What are the typical exam questions based on velocity of object and image in CBSE syllabus?
Common CBSE exam questions on this topic focus on applying formulas, sign convention, and conceptual understanding. Typical patterns include:
- Calculate the velocity of the image given object’s velocity and position for mirrors/lenses.
- Reasoning-based questions on why image velocity is greater or less than object velocity.
- Numerical problems with concave/convex mirror setups.
- Application to daily life situations (e.g., rearview mirrors).
7. Can we use the same formula for velocity of images in both mirrors and lenses?
Yes, the same differentiated equation applies to both mirrors and lenses with appropriate adjustments for sign conventions. The process involves:
- Using the respective formula (mirror: 1/f = 1/v + 1/u, lens: 1/f = 1/v - 1/u).
- Differentiating w.r.t time.
- Applying proper positive/negative signs for object and image distances.
8. Why does the velocity of the image sometimes exceed the velocity of the object?
The velocity of the image can be greater than the object due to the geometrical properties of mirrors and magnification effects.
- For concave mirrors (object near focus), small object movements produce large image displacements.
- The mathematical relationship (v_i = (v_o * v^2)/u^2) shows possible amplification.
- This is called ‘lateral magnification’ in optics.
9. What is the formula for the velocity of image formed by a concave mirror if the object moves with speed v towards the mirror?
The velocity of the image (v_i) for a concave mirror can be mathematically expressed as:
- First, use the mirror equation: 1/f = 1/v + 1/u
- Differentiating: 0 = -1/v2(dv/dt) - 1/u2(du/dt)
- Solving for dv/dt: dv/dt = - (v2/u2) (du/dt)
- If the object moves towards the mirror, (du/dt) is negative (as object distance decreases).
10. How does magnification relate to the velocity of image in optics?
Magnification in optics directly affects how much the velocity of the image differs from the object.
- Magnification (m) = Image height / Object height = v/u.
- The velocity of image is tied to this by (v_i/v_o) = (v^2/u^2).
- Larger magnification for specific placements (such as near the focal point) leads to higher image velocities.
