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In this article, you will learn what we mean by **instantaneous velocity** when describing the motion of a particle.

The instantaneous velocity v that a particle has at an instant t is equal to the value that the average velocity, calculated over an interval of time Î”t which includes t, approaches as Î”t approaches 0. So, the instantaneous velocity v is expressed as:

v = | lim | Î”x | |

Î”tâ†’0 | Î”t |

The interval of time Î”t starts at some instant t_{1} and ends at some subsequent instant t_{2}, so we can write:

Î”t = t_{2} âˆ’ t_{1}

And since the instant t is within this interval of time, it follows that:

t_{1} â‰¤ t â‰¤ t_{2}

To better see what we're talking about, let's represent this on a position vs time graph of a particle:

The arrows next to t_{1} and t_{2} are there to indicate that we are considering the interval of time Î”t to get smaller and smaller.

Now, since t_{1} â‰¤ t â‰¤ t_{2}, let's consider t_{1} = t. When we do this, as Î”t approaches 0, t_{2} gets closer and closer to t_{1}:

Since Î”t = t_{2} âˆ’ t_{1} and t_{1} = t, we can write:

Î”t = t_{2} âˆ’ t

â†“

t_{2} = t + Î”t

This means that the extremes of Î”t are t and t + Î”t:

Since the instantaneous velocity at t is equal to whatever the average velocity for Î”t approaches as Î”t approaches 0:

v = | lim | Î”x | |

Î”tâ†’0 | Î”t |

and we know that the change in position Î”x is

Î”x = x_{t+Î”t} âˆ’ x_{t}

it follows that we can write the instantaneous velocity v as

v = | lim | x_{t+Î”t} âˆ’ x_{t} | |

Î”tâ†’0 | Î”t |

Let's now go through an example to demonstrate how to use this formula.

Let's consider an object whose position in meters at an instant t, specified in seconds, is given by t^{2}:

x = t^{2}

So, at 1 s the position is 1 m, at 2 s the position is 4 m, at 3 s the position is 9 m, etc.

Our goal here is to find what the instantaneous velocity of the object is at any instant t.

As we've seen, the instantaneous velocity v at an instant t is given by:

v = | lim | x_{t+Î”t} âˆ’ x_{t} | |

Î”tâ†’0 | Î”t |

Since in this case the position of the object at any instant t is given by t^{2}, the positions at the instants t and t + Î”t will be

x_{t} = t^{2}

x_{t+Î”t} = (t + Î”t)^{2}

Thus, the instantaneous velocity v becomes:

v = | lim | (t + Î”t)^{2} âˆ’ t^{2} | |

Î”tâ†’0 | Î”t |

Let's work out the numerator:

v = | lim | t^{2} + 2tÎ”t + Î”t^{2} âˆ’ t^{2} | |

Î”tâ†’0 | Î”t |

v = | lim | 2tÎ”t + Î”t^{2} | |

Î”tâ†’0 | Î”t |

Next, we can divide the numerator by Î”t:

v = | lim | 2t + Î”t |

Î”tâ†’0 |

As Î”t approaches 0, 2t + Î”t approaches 2t, so we can write:

v = 2t

Therefore, the instantaneous velocity of the object at any instant t is given by 2t. So, at 1 s the instantaneous velocity is 2 m/s, at 2 s the instantaneous velocity is 4 m/s, at 3 s the instantaneous velocity is 6 m/s, etc.

The limit

lim | x_{t+Î”t} âˆ’ x_{t} | |

Î”tâ†’0 | Î”t |

is called the derivative of x with respect to t, and is indicated as

dx |

dt |

So, we can write

v = | lim | x_{t+Î”t} âˆ’ x_{t} | = | dx | |

Î”tâ†’0 | Î”t | dt |

Therefore, we say that the instantaneous velocity is the derivative of position with respect to time.

Let's return to the position vs time graph we've seen before and let's draw a secant line passing through the points t and t + Î”t:

As explained in the average velocity article, the slope of this secant line is equal to the average velocity for Î”t because Î”x/Î”t is at the same time the average velocity and the slope of the secant line.

Since the instantaneous velocity at t is what the average velocity for Î”t approaches as Î”t approaches 0, it follows that the instantaneous velocity at t is what the slope of the secant line approaches as Î”t approaches 0.

As Î”t gets smaller and smaller, the secant line gets closer and closer to the line tangent to the graph at the point t:

So, as Î”t approaches 0, the slope of the secant line approaches the slope of the line tangent to the graph at the point t.

Therefore, the instantaneous velocity at t is equal to the slope of the line tangent to the graph at the point t.

If we indicate the slope of the tangent line with m_{T}, we can write

v = | dx | = m_{T} |

dt |

Let's call Î¸ the angle that the line tangent to the graph at the point t makes with the positive t-axis:

We can look at the sign of the angle Î¸ to determine whether the instantaneous velocity is positive, negative, or zero:

- if Î¸ > 0 â†’ m
_{T}> 0 â†’ v > 0 - if Î¸ < 0 â†’ m
_{T}< 0 â†’ v < 0 - if Î¸ = 0 â†’ m
_{T}= 0 â†’ v = 0

Thus, we can easily determine when the velocity is positive, negative, and zero, simply by looking at the angle Î¸ at different points on a position vs time graph: