How can you find the velocity at a particular time?

Question

I was studying about instantaneous velocity but at a particular time the change in position would be zero and also the change in time is also zero . So what are we finding?

Answer 1

Velocity is displacement $\Delta s$ (change in position) over duration $\Delta t$ (change in time):

$$v=\frac{\Delta s}{\Delta t}.$$

• If $\Delta s=0$ then there is no speed, $v=0$.
• If $\Delta t=0$ then the speed is enormous, $v\to \infty$.

But what if both are zero?

In fact, neither is exactly zero, but just very, very close to zero. So close that they cover just a moment. We typically change the notation to:

$$v=\frac{\mathrm ds}{\mathrm dt}$$

to indicate such momentary, negligible, infinitesimal, basically instantaneous changes. When the duration is shortened down to become very, very small we say mathematically that it tends towards or goes towards zero, $\mathrm dt\to 0$. Becaue the displacement follows the duration, we simultaneously see $\mathrm ds\to 0$. If just one of them tended towards zero, then we could deduce what $v$ tends to as above. But when both tend towards zero, then their fraction has a value. That value is the speed and it applies over that brief moment.

So, to round it up: we are actually not finding the speed as zero displacement over zero time, but rather for values that tend towards zero. And that makes all the difference. Mathematically you can't divide by zero, but you can find the so-called limit where the denominator goes towards zero. This very important realisation in math brought to the invention of calculus which changed the physical world entirely, making what we today consider a simple task, such as defining instantaneous velocity, possible.

Answer 2

If you know nothing of some moving object but its position at one specific time you can not determine its velocity. If you know its position s over time so you know s(t) you can define s'(t1) as the instantaneous velocity at time t1 since if you measure over very little time intervalls you approach this velocity. If you do not know about differentiation , this is too difficult and you just think of very short time intervalls an distances and calculate the velocity in this short times.

Answer 3

At a particular time, the change in position is not zero. For example, suppose a body covers a distance of 10 m on moving from position A to position B. At time 2 s, we have to find the instantaneous change in position of object at time t = 2 s. Then tell me, my dear, at what instant do you calculate this?