1. What is Instantaneous Velocity?

Instantaneous velocity is the velocity of an object at a specific instant in time. It is defined as the derivative of the position function with respect to time, representing the rate of change of position at that exact moment.

2. How Does the Calculator Work?

The calculator uses the fundamental calculus concept:

Where:

• \( v \) — Instantaneous velocity
• \( s \) — Position function
• \( t \) — Time
• \( \frac{ds}{dt} \) — Derivative of position with respect to time

Explanation: The calculator computes the derivative of the given position function and evaluates it at the specified time to find the instantaneous velocity.

3. Importance of Instantaneous Velocity

Details: Instantaneous velocity is crucial in physics and engineering for analyzing motion, predicting trajectories, and understanding the behavior of moving objects at specific moments in time.

4. Using the Calculator

Tips: Enter the position function as a mathematical expression in terms of 't', and specify the time at which you want to calculate the instantaneous velocity.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between average and instantaneous velocity?
A: Average velocity is the total displacement divided by total time, while instantaneous velocity is the velocity at a specific instant.

Q2: Can I use any position function?
A: The calculator should work with differentiable functions. Complex functions may require specialized mathematical libraries.

Q3: What are common position functions?
A: Common examples include polynomials (t², t³), trigonometric functions (sin(t), cos(t)), and exponential functions.

Q4: How is this different from speed?
A: Velocity includes direction (vector quantity), while speed is the magnitude of velocity (scalar quantity).

Q5: What if the derivative doesn't exist at a point?
A: If the position function is not differentiable at the given time, the instantaneous velocity is undefined at that point.