ANAND CLASSES study material and notes which explore the concept of Instantaneous speed. Instantaneous speed is the speed of an object at a specific moment in time. Learn its definition, formula, difference from average speed, calculations, examples, FAQs, MCQs, and conceptual questions for JEE, NEET, and CBSE Class 11 exams.
Understanding Instantaneous Speed
Instantaneous speed is a fundamental concept in physics, representing the speed of an object at a specific moment in time. Unlike average speed, which considers the total distance traveled over a period, instantaneous speed focuses on an exact point in time. This concept is crucial for students preparing for exams like JEE, NEET, and CBSE Class 11, as it lays the foundation for more advanced topics in mechanics.
Definition of Instantaneous Speed
Instantaneous speed is defined as the magnitude of the instantaneous velocity of an object at a particular instant.
It indicates how fast an object is moving at a specific moment, without considering the direction of motion.
Mathematically,
If $s(t)$ represents the position of an object as a function of time $t$, the instantaneous speed $v$ at time tt is given by the derivative of $s(t)$ with respect to $t$ :
$$v_i = \left| \frac{ds(t)}{dt} \right|$$
This derivative represents the rate of change of position with respect to time, providing the speed at that specific instant.
Difference Between Instantaneous Speed and Average Speed
It’s essential to distinguish between instantaneous speed and average speed :
- Instantaneous Speed: The speed of an object at a specific moment in time.
- Average Speed: The total distance traveled divided by the total time taken.
For example, if a car travels 100 kilometers in 2 hours, its average speed is 50 km/h. However, during the trip, the car’s instantaneous speed may vary, sometimes exceeding or falling below the average speed.
Below is a detailed comparison :
| Feature | Instantaneous Speed | Average Speed |
|---|---|---|
| Definition | The speed of an object at a specific instant of time. | The total distance traveled divided by the total time taken. |
| Formula | $$v_i = \left| \frac{ds(t)}{dt} \right|$$ | $$v_{av} = \frac{\text{Total Distance Travelled}}{\text{Total Time Taken}}$$ |
| Time Consideration | Measured at a single instant. | Measured over a time interval. |
| Variation | Can change from moment to moment. | Represents the overall motion over a period of time. |
| Example | The reading on a car’s speedometer at a given instant. | The average speed of a car over a long journey. |
| Measurement Tool | Speedometer, derivative of position-time function. | Total distance and time measurement. |
Key Takeaways:
- Instantaneous speed is like checking your car’s speedometer at a particular second.
- Average speed is like calculating your trip’s speed by dividing total distance by total time.
- In cases of constant speed, instantaneous speed = average speed.
- When an object’s speed varies, instantaneous speed fluctuates, while average speed provides an overall measure.
This distinction is crucial in physics, particularly in kinematics, where motion is analyzed mathematically and graphically.
Calculating Instantaneous Speed
To calculate instantaneous speed, follow these steps:
- Determine the Position Function: Identify the function $s(t)$ that describes the object’s position over time.
- Differentiate the Position Function: Compute the derivative of $s(t)$ with respect to time tt to obtain the velocity function $v(t)$.
- Evaluate the Magnitude: Take the absolute value of $v(t)$ at the desired time to find the instantaneous speed.
Example:
Consider an object whose position is given by $$s(t) = 5t^2 + 3t + 2$$
To find the instantaneous speed at t = 2 seconds :
- Differentiate: $$v(t) = \frac{d}{dt}(5t^2 + 3t + 2) = 10t + 3$$
- Evaluate: $$v(2) = 10(2) + 3 = 23 m/s$$
- Instantaneous Speed: $$|v(2)| = |23| = 23 m/s$$
Importance in Physics and Examinations
Understanding instantaneous speed is vital for analyzing real-world scenarios where speed varies over time, such as accelerating vehicles or objects in free fall. In examinations like JEE and NEET, questions often test the ability to apply calculus concepts to determine instantaneous speed, making it a critical topic for students.
Frequently Asked Questions (FAQs)
Q1: How does instantaneous speed differ from instantaneous velocity?
A1: Instantaneous speed is the magnitude of instantaneous velocity and does not include direction, whereas instantaneous velocity is a vector quantity that includes both magnitude and direction.
Q2: Can instantaneous speed be negative?
A2: No, instantaneous speed, being the magnitude of velocity, is always a non-negative value.
Q3: How is instantaneous speed measured in real-life scenarios?
A3: Devices like speedometers in vehicles measure instantaneous speed by calculating the rate of change of position over very short time intervals.
Q4: Why is understanding instantaneous speed important for competitive exams?
A4: Many problems in competitive exams involve analyzing motion at specific instants, requiring a solid understanding of instantaneous speed and its calculation.
Multiple Choice Questions (MCQs)
Q1: If the position of a particle is given by $s(t) = 4t^3 – 2t + 1$, what is its instantaneous speed at $\:t = 1 \:second $ ?
A. 10 m/s
B. 12 m/s
C. 14 m/s
D. 16 m/s
Answer: B. 12 m/s
Explanation:
- Differentiate $s(t)$ : $$v(t) = \frac{d}{dt}(4t^3 – 2t + 1) = 12t^2 – 2$$
- Evaluate at $t = 1$ : $$v(1) = 12(1)^2 – 2 = 10 m/s$$
- Instantaneous speed: $$|v(1)| = |10| = 10 m/s$$
Q2: A car’s velocity is described by $v(t) = 3t^2 + 2t$. What is its instantaneous speed at $ t = 2 \:seconds$ ?
A. 16 m/s
B. 18 m/s
C. 20 m/s
D. 22 m/s
Answer: C. 20 m/s
Explanation:
- Instantaneous speed at $t = 2$ : $$v(2) = 3(2)^2 + 2(2) = 12 + 4 = 16\: m/s$$
- Instantaneous speed: $$|v(2)| = |16| = 16 \:m/s$$
Conceptual Questions with Answers
Q1: Why is the concept of instantaneous speed necessary when we already have average speed?
Answer: Average speed provides information over a time interval but doesn’t capture variations within time interval.
