To define what it means to say that a car is going at the speed of 2mph would again require the use of limits. If $\lim_{h\to 0}\frac{x(t_0+h)-x(t_0)}{h}=2$ then the speed at $t=t_0$ is $2$, where $x(t)$ is the displacement of the car from any given reference at time $t$. But I guess you already knew that. I don't what more can I say.Poirot said:I'm not really sure how this concept makes sense. We give it meaning by way of a limit but, to my mind, at any instant the car is motionless so how can it have a velocity? What, in essence, does it mean to say that at some point in time a car is going 2 mph?
Poirot said:At time t, the car is stationary...
Would you agree that the 'average rate of change' between $2$ and $4$ of the function $y=x^2$ is $(16-4)/2=6$?Poirot said:In fact, the 'issue' generalises to differentiation of non-linear functions. Take f(x)=x^2.
when x=2, the derivative is 2. Now if one is asked to explain what a derivative tells you, then you would probably say along the lines 'the rate of change of y with x'. But what does the rate of change mean at a single point? I hope this helps you to understand.
It means exactly what the definition says. The rate of change at a point is the limit of the average rate of change over intervals containing the point as the length of the intervals decrease to zero (with due allowances for end effects which I can't be bothered with here).Poirot said:In fact, the 'issue' generalises to differentiation of non-linear functions. Take f(x)=x^2.
when x=2, the derivative is 2. Now if one is asked to explain what a derivative tells you, then you would probably say along the lines 'the rate of change of y with x'. But what does the rate of change mean at a single point? I hope this helps you to understand.
Instantaneous velocity is the rate of change of an object's position at a specific moment in time. It is the slope of the object's position-time graph at that particular point.
Average velocity is the total displacement of an object divided by the total time taken. It gives an overall idea of an object's motion. On the other hand, instantaneous velocity is the velocity at a particular instant and can vary throughout an object's motion.
To calculate instantaneous velocity, we use the formula v = Δx/Δt, where v is the velocity, Δx is the change in position, and Δt is the change in time. We can also use the derivative of the position function with respect to time to calculate instantaneous velocity.
Yes, instantaneous velocity can be positive, zero, or negative, depending on the direction of an object's motion. For example, if an object is moving in the positive direction, its instantaneous velocity will be positive, and if it is moving in the negative direction, its instantaneous velocity will be negative.
Instantaneous velocity is crucial in physics because it helps us understand the motion of an object at a particular moment. It allows us to analyze an object's acceleration, direction of motion, and position at a specific time. It also helps in determining the forces acting on an object and predicting its future motion.