What is the derivative definition of instantaneous velocity?

Answer 1

Instantaneous velocity is the change in position over the change in time. Therefore, the derivative definition of instantaneous velocity is:

instantaneous velocity= #v#= #lim_(Delta t -> 0) ##(Delta x)/(Delta t)#= #dx/dt#

So basically, instantaneous velocity is the derivative of the position function/equation of motion. For example, let's say you had a position function:

#x=6t^2+t+12#
Since #v#=#dx/dt#, #v= d/dt 6t^2+t+12= 12t+1#
That is the function of the instantaneous velocity in this case. Note that it is a function because instantaneous velocity is variable- It is dependent on time, or the "instant." For every #t#, there is a different velocity at that given instant #t#.
Let's say we wanted to know the velocity at #t=10# and the position is measured in meters (m) while the time in measured in seconds (sec).
#v=12(10)+1= 121 (m)/(sec)#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

The derivative definition of instantaneous velocity is the rate of change of position with respect to time at a specific instant. Mathematically, it is expressed as the derivative of the position function ( s(t) ) with respect to time ( t ) at a particular time ( t_0 ). In symbols, the instantaneous velocity ( v(t_0) ) at time ( t_0 ) is given by:

[ v(t_0) = \lim_{\Delta t \to 0} \frac{s(t_0 + \Delta t) - s(t_0)}{\Delta t} ]

Where:

  • ( v(t_0) ) is the instantaneous velocity at time ( t_0 ),
  • ( s(t) ) is the position function representing the displacement of an object at time ( t ),
  • ( \Delta t ) is a small interval of time,
  • ( s(t_0 + \Delta t) - s(t_0) ) represents the change in position over the time interval ( \Delta t ),
  • The limit as ( \Delta t ) approaches 0 ensures that the velocity is calculated at an infinitesimally small time interval around ( t_0 ), capturing the instantaneous rate of change of position with respect to time.
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7