ANAND CLASSES study material and notes to learn about Instantaneous Velocity with its definition, formula, and key differences from instantaneous speed and average speed. Get clear explanations with equations and FAQs.
Definition of Instantaneous Velocity
Instantaneous velocity is defined as the rate of change of the position vector of a particle with respect to time at a certain instant. Mathematically, it is given by:
$$v_i = \lim_{\Delta t \to 0} \frac{\Delta \vec{r}}{\Delta t} = \frac{d\vec{r}}{dt}$$
This means that instantaneous velocity is the derivative of displacement with respect to time and gives the velocity of a particle at a specific moment.
Difference Between Instantaneous Speed and Instantaneous Velocity
Both instantaneous velocity and average velocity describe an object’s motion, but they differ in their calculation and interpretation.
(a) Instantaneous velocity is always tangential to the path of the particle.
- Example: When a stone is thrown from a point O, the instantaneous velocity at different points along the trajectory is:
- $v_1$ at the point of projection O
- $v_2$ at point A
- $v_3$ at point B
- $v_4$ at point C
- The direction of these velocities can be determined by drawing a tangent at each point on the trajectory.
(b) A particle may have constant instantaneous speed but variable instantaneous velocity.
- Example: A particle undergoing uniform circular motion maintains constant speed, but its velocity changes at every instant because of continuous change in direction.
(c) Magnitude of instantaneous velocity = Instantaneous speed
- The numerical value of instantaneous velocity (ignoring direction) is the instantaneous speed of the particle.
(d) For motion with constant velocity, instantaneous velocity = average velocity.
- When a particle moves with uniform velocity, the instantaneous velocity remains equal to its average velocity over any time interval.
(e) Derivation of instantaneous velocity from displacement function
If the displacement of a particle is given by: $$x = A_2t^2 + A_1t + A_0$$ Then, differentiating with respect to time: $$v = \frac{dx}{dt} = 2A_2t + A_1 $$ For $$t = 0$$
- Instantaneous velocity: $$v_i = A_1$$
- Instantaneous speed: $$v_i =|A_1|$$
Key Differences
| Feature | Instantaneous Velocity | Average Velocity |
|---|---|---|
| Definition | Velocity at a specific instant of time. | Displacement per unit time over an interval. |
| Formula | $v= \frac{dx}{dt}$(Derivative of displacement) | $v = \frac{\text{Total Displacement}}{\text{Total Time}}$ |
| Measurement | Measured using calculus or a speedometer at a moment. | Computed from initial and final positions. |
| Variation | Can change at every instant. | Represents an overall trend in motion. |
| Depends On | Instantaneous rate of change of position. | Net displacement, not total distance. |
| Example | Speed and direction of a car at exactly 5 seconds. | Car’s average velocity over a 2-hour journey. |
📌 Key Takeaways
- Instantaneous velocity shows how fast and in what direction an object moves at a particular moment.
- Average velocity gives the overall change in position per unit time, ignoring variations in motion.
- If an object changes direction, instantaneous velocity varies, but average velocity depends only on net displacement.
📌 Example:
- A car moving from A to B and back to A in 2 hours has zero average velocity (displacement = 0), but its instantaneous velocity was nonzero throughout the motion.
FAQs on Instantaneous Velocity
Q1: Is instantaneous velocity always a vector?
Yes, instantaneous velocity has both magnitude and direction, making it a vector quantity.
Q2: Can instantaneous velocity be negative?
Yes, if the particle is moving in the opposite direction to the chosen reference direction, its instantaneous velocity will be negative.
Q3: How is instantaneous velocity measured practically?
It is measured using speedometers, motion sensors, or by differentiating the position-time function.
Q4: What is the instantaneous velocity of a freely falling object just before hitting the ground?
Using $v = u + gt$, where $u = 0$ (for an object dropped), instantaneous velocity just before impact is $v = gt$.
Multiple-Choice Questions (MCQs)
Q1: Which of the following is true about instantaneous velocity?
(a) It is always in the direction of acceleration.
(b) It is always in the direction of displacement.
(c) It is tangent to the path of motion.
(d) It is always constant.
Answer: (c) Instantaneous velocity is always tangential to the path.
Q2: If a particle moves in a straight line with uniform velocity, its instantaneous velocity at any instant is
(a) Equal to the average velocity
(b) Less than the average velocity
(c) More than the average velocity
(d) Zero
Answer: (a) Because in uniform motion, instantaneous and average velocity are the same.
Conceptual Questions
Q1: A particle moves in a circular path with constant speed. What can be said about its instantaneous velocity?
- It is continuously changing direction but has a constant magnitude.
Q2: Can instantaneous velocity ever be greater than average velocity?
- Yes, if the particle’s velocity varies significantly over time.
Do You Know?
- Instantaneous velocity is the derivative of the displacement function, while instantaneous acceleration is the derivative of the velocity function.
- The concept of instantaneous velocity is essential in physics, especially in kinematics and calculus-based mechanics.
- GPS systems and modern motion detectors use instantaneous velocity concepts for navigation.
Worksheet on Instantaneous Velocity
Question 1: If $x = 4t^2 + 3t – 5$, find instantaneous velocity at $t = 2s$.
Question 2: A body moves such that its velocity at any instant is given by $v = 5t + 2$. Find its displacement in the first 4 seconds.
Test Paper on Instantaneous Velocity (Total: 20 Marks)
- Define instantaneous velocity. (2 marks)
- Differentiate between instantaneous velocity and instantaneous speed. (3 marks)
- A particle moves along a straight line such that its displacement $x = 3t^2 – 2t + 5$. Find its instantaneous velocity at $t = 4 \:seconds$. (5 marks)
- Derive an expression for instantaneous velocity in terms of displacement function. (5 marks)
- A car moves with velocity $v = 10 + 2t$ m/s. Find its velocity at $t = 3\: seconds$. (5 marks)
Important Points for Quick Revision
- Instantaneous velocity is the derivative of displacement.
- It is tangential to the path of motion.
- A particle with constant speed can have a changing instantaneous velocity.
- In uniform motion, instantaneous velocity equals average velocity.
- Measured using speedometers, sensors, or differentiation.
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