What is the formula of instantaneous speed?

Question

don't know anything about it. except the definition. which is the rate at which an object is moving at a given moment in time. by the way what does df(t) mean?

Answer 1

If the distance 'x' travelled is a function of time, like this:

x = f(t),

then instantaneous speed (or rate at which the object is moving), v(t), is given by:

v(t) = dx/dt = d f(t)/dt; and

instantaneous acceleration (rate of change of v(t)), a(t), is given by:

a(t) = d [v(t)]/dt = d^2 x(t)/dt^2 = d^2 f(t)/dt^2.

All the "d's" are calculus operations, and a verbal equivalent of writing the "operator" d/dt is to say "the derivative of ...*** with respect to the independent variable t." (You can see how much more compact the math is!)

[*** Here, ... means the function or variable being operated on.]

The "operator" d .../dt always gets you the INSTANTANEOUS time rate of change of whatever it's applied to (provided that 't' means 'time'!)

d^2 ... /dt^2 means "apply the differentiating operator twice."

If you haven't met Calculus yet, this won't mean anything to you. However, you essentially asked a Calculus question, and df(t) is part of df(t)/dt, the derivative of f(t) with respect to t.

Live long and prosper.

Answer 2

If you have an equation for the speed, just plug in the coordinate of where you want the instantaneous speed at and that's the answer. If you have an equation for location as a function of time, you need to take the first derivative and plug in the time. If you're equation for speed is x^2 +x + 4, and you want the instantaneous speed at x = 6, just plug in 6 for x and that's your answer. If your equation for x as a function of time is, 2 * t^2, take the 1st derivative which is 4*t and plug in the value for t you're interested in and you have the answer

Answer 3

If you have a formula for the position of an object at time t:

The first derivative is the velocity at time t
The second derivative is the acceleration at time t

I think you are talking about the velocity of an object at a particular time.