The instantaneous velocity of a particle is equal to time derivative of a position vector and instantaneous acceleration is equal to time derivative of its velocity vector then why instantaneous acceleration is independent of instantaneous position vector and instantaneous velocity?

Question

Answer

Explanation:

Instantaneous velocity: The instantaneous velocity of a particle is equal to the time derivative of its position vector. It means the rate of change of position with respect to time at a particular instant of time.

Instantaneous acceleration: The instantaneous acceleration of a particle is equal to the time derivative of its velocity vector. It means the rate of change of velocity with respect to time at a particular instant of time.

Independence of instantaneous acceleration: The instantaneous acceleration is independent of instantaneous position vector and instantaneous velocity because:

The instantaneous acceleration of a particle depends only on the rate of change of velocity with respect to time, not on the position of the particle.

Even if the position of the particle changes at a particular instant of time, it doesn't affect the instantaneous acceleration of the particle at that instant of time.

Similarly, the instantaneous acceleration of a particle also doesn't depend on its instantaneous velocity because the rate of change of velocity with respect to time is independent of the magnitude and direction of velocity.

Example:
Suppose a car is moving with a constant velocity of 40 km/hr and suddenly accelerates to 60 km/hr in 5 seconds. Here, the instantaneous velocity of the car changes from 40 km/hr to 60 km/hr in 5 seconds, which means the car experiences an instantaneous acceleration of 4 km/hr/s. This instantaneous acceleration doesn't depend on the position of the car or its velocity; it only depends on the rate of change of velocity with respect to time.

Answer

When a car is driving down the freeway, at any given point in time it has a certain exact velocity. This is the true velocity of the car at that instant, i.e. instantaneous velocity.

We define velocity as the rate of change in displacement, divided by the rate of change in time:

v = ∆ s / ∆ t

Where

v

is velocity,

∆ s

is the change in displacement, and

∆ t

is the change in time.

For example, in 6 seconds a car travels 12 meters, so it's velocity is

v = 12 / 6 = 2

m / s

Now in this example, what we've found is actually the car's average velocity. In reality, the car may have been stationary for 5 seconds, and then for the last one second travelled at

12 m / s

.

Over a 6 second period, it has still travelled 12 meters, so when we plug those numbers into our equation, we're still going to get

2 m / s

So this is average velocity.

In order to find instantaneous velocity, we simple change our time period to 0 seconds. So we're no longer finding the average velocity over a period of time, but rather the velocity at an exact instant of time.

v = ∆ s / ∆ t = ∆ s / 0

However in reality it's not that easy. Instantaneous velocity is difficult to calculate because it causes us to divide by zero.

In the real word we either approximate it by making

∆ t

very small. Or we use fancy mathematics involving limits and equations of motion in order to work out instantaneous velocity.

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