show that accelration at any instant does not depend upon velocity at any instant?
To show that acceleration at any instant does not depend on velocity at any instant, we can use the definition of acceleration and the basic principles of calculus.
Acceleration is defined as the rate of change of velocity with respect to time. Mathematically, this can be expressed as:
a = dv/dt
Where "a" represents acceleration, "v" represents velocity, and "dt" represents the change in time.
Now, let's assume that acceleration does depend on velocity at any instant, which means that a = f(v), where "f" is some function of velocity.
To prove that this assumption is false, we need to show that the derivative of acceleration with respect to time does not involve the derivative of velocity with respect to time.
Taking the derivative of both sides of the equation a = f(v) with respect to time, we get:
da/dt = d/dt(f(v))
Using the chain rule of calculus, we can express the derivative of a function of a function as:
da/dt = df/dv * dv/dt
Now, if acceleration depends on velocity at any instant, then df/dv would not be zero, and we can rewrite the equation as:
da/dt = df/dv * dv/dt ≠ 0
This would mean that the acceleration is changing with respect to time, which contradicts our initial assumption that acceleration does not depend on velocity at any instant.
Hence, we can conclude that acceleration at any instant does not depend on velocity at any instant.