We already know what is speed in physics. In this article, we will learn about instantaneous speed. We would also look at a few example questions based on instantaneous speed.

**Speed Definition**

Speed is defined as the rate at which the position of any object changes over time. An objectâ€™s speed might vary as it moves.

## Instantaneous Speed

### Definition

Instantaneous speedÂ is the speed of a moving object at any particular instant of time.

It is a term used to describe the speed of an object at any one point in time. The instantaneous speed of a moving objectÂ might vary from one moment to another.

A movingÂ carâ€™s speedometer, for example, displays the instantaneous speed at which the car is moving. At one instant of time, the carâ€™s instantaneous speed maybe 60 mph, and in the very next moment, applying a tiny push force on the accelerator pedal may cause the instantaneous speed to increase to 66 mph.

## Instantaneous Speed Formula

An objectâ€™s instantaneous speed can also be described as a short distance travelled divided by the time it takes to cover that distance.

The instantaneous speed formula is given by the relation

$$\text{Instantaneous Speed, }v_i=\lim_{\Delta t\rightarrow 0}\frac{s(t+\Delta t)-s(t)}{\Delta t}$$

or,

$$v_i=\lim_{\Delta t\rightarrow 0}\frac{\Delta s}{\Delta t}=\frac{ds}{dt}$$

where,

$v_i$ is the instantaneous speed

$\Delta s$ is change in value of $s$

$\Delta t$ is change in time $t$

The figure given below shows a distance-time graph. The slope of the tangent to the graph at time $t$ gives the speed at time $t$. The slope of chord $AB$, as shown in the distance-time graph below, gives the average speed between $t$ and $t+\Delta t$. $A$ and $B$ are the points on the curve corresponding to time $t$ and $t+\Delta t$, respectively.

If $\Delta t \rightarrow 0$, the chord $AB$ becomes the tangent to the curve at the point $A$, and the average speed $\frac{\Delta s}{\Delta t}$ becomes the slope of the tangent and is expressed as $\frac{ds}{dt}$.

Here, $ds$ is the small distance covered and $dt$ is the small time taken in travelling the same distance.

### Important Points To Remember

- The instantaneous speed is never less than or equal to zero.
- It is a scalar quantity.
- The instantaneous speed of uniform motion is constant.
- The magnitude of instantaneous velocity at any given time is the magnitude of instantaneous speed at that time.Â
- It is a limit taken over the average speed when the time interval becomes very small or approaches zero.

## How to Calculate Instantaneous Speed

Now we know about the instantaneous speed definition and formula, let us now look at some example problems on how to calculate it.

**Example Question:-** When an object is dropped from a particular height, it begins to fall under the influence of gravity. The objectâ€™s position changes in accordance with the function $s(t)=10t^2$, where $s(t)$ is measured in meters. Calculate the objectâ€™s instantaneous speed at $t=15$ seconds.

**Solution:-**

We know that instantaneous speed is given by the relation

$v_i=\lim_{\Delta t\rightarrow 0}\frac{\Delta s}{\Delta t}=\frac{ds}{dt}$

It is given in the question that

$s(t)=10t^2$,

We can find instantaneous velocity by differentiating $s(t)$ with respect to time.

$\Rightarrow v_i=(\frac{ds}{dt}){t=15}$

Now,

$\frac{ds}{dt}=\frac{d}{dt}(10t^2)=2\times 10 \times t$

So,

$\Rightarrow v_i=(\frac{ds}{dt}){t=15}=2\times 10 \times 15 = 300 m/s$

The instantaneous speed at $t=10 s$ is $300 m/s$