KAHIBARO
Table of Contents
Seeing motion at a single instant
When an object moves along a straight line, its position changes with time. Average velocity tells us how fast position changes over a time interval, but sometimes we want something more precise. We want to know how the object is moving at one exact moment. That is the idea of instantaneous velocity.
Instantaneous velocity is the velocity of an object at a particular instant of time. It tells us both how fast the object is moving and in which direction along the line it is moving at that moment.
If the object is moving in the positive direction, the instantaneous velocity is positive. If it is moving in the negative direction, the instantaneous velocity is negative. If it is momentarily at rest, the instantaneous velocity is zero.
From average velocity to instantaneous velocity
Suppose the position of an object is $x(t)$. Over a small time interval $\Delta t$, the displacement is $\Delta x = x(t + \Delta t) - x(t)$. The average velocity over that interval is
$$
v_{\text{avg}} = \frac{\Delta x}{\Delta t}
$$
To get the instantaneous velocity, we make the time interval smaller and smaller. In the limit as $\Delta t$ approaches zero, average velocity becomes instantaneous velocity:
$$
v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}
$$
This is the derivative of position with respect to time:
$$
v = \frac{dx}{dt}
$$
Instantaneous velocity is defined by
$$
v = \frac{dx}{dt}
$$
It is the rate of change of position with time at a single instant.
Meaning of the sign
Because motion in one dimension happens along a line, the sign of velocity matters a lot. A positive velocity means the object moves in the positive direction of the chosen axis. A negative velocity means it moves in the opposite direction.
This is why velocity is different from speed. Speed only tells how fast something moves. Velocity tells both magnitude and direction.
Here is a simple comparison:
| Quantity | Can be positive or negative? | Includes direction? |
|---|---|---|
| Speed | No | No |
| Velocity | Yes | Yes |
For example, if a car moves to the right and we choose right as positive, then $v = +12\ \text{m/s}$ means the car moves right at $12\ \text{m/s}$. If $v = -12\ \text{m/s}$, it moves left at the same rate.
Instantaneous velocity on a graph
A position versus time graph is one of the best ways to understand instantaneous velocity. At any point on the graph, the instantaneous velocity is the slope of the tangent line to the curve at that point.
If the tangent line slopes upward, the velocity is positive. If it slopes downward, the velocity is negative. If the tangent line is horizontal, the velocity is zero.
On a position-time graph, instantaneous velocity equals the slope of the tangent line:
$$
v = \frac{dx}{dt}
$$
A steeper tangent means a larger magnitude of velocity.
Instantaneous velocity from an equation
If the position is given as a function of time, we can find instantaneous velocity by differentiating.
Suppose
$$
x(t) = 4t^2 + 2t
$$
Then
$$
v(t) = \frac{dx}{dt} = 8t + 2
$$
This means the velocity depends on time. At different moments, the object has different instantaneous velocities.
At $t = 1\ \text{s}$,
$$
v(1) = 8(1) + 2 = 10\ \text{m/s}
$$
At $t = 3\ \text{s}$,
$$
v(3) = 8(3) + 2 = 26\ \text{m/s}
$$
So the object is moving faster later in time.
A simple example with changing direction
Consider
$$
x(t) = t^2 - 4t
$$
Then
$$
v(t) = \frac{dx}{dt} = 2t - 4
$$
Now look at a few times:
| Time $t$ | Instantaneous velocity $v(t)$ | Meaning |
|---|---|---|
| $0$ | $-4$ | Moving in negative direction |
| $2$ | $0$ | Momentarily at rest |
| $3$ | $2$ | Moving in positive direction |
At $t = 2\ \text{s}$, the velocity is zero. This does not necessarily mean the motion is finished. It may mean the object stops for an instant and then reverses direction.
If $v = 0$ at an instant, the object is momentarily at rest at that instant. It may still start moving again immediately after.
Units of instantaneous velocity
The SI unit of instantaneous velocity is meters per second, written as $\text{m/s}$. Since velocity is position divided by time,
$$
\frac{\text{meter}}{\text{second}} = \text{m/s}
$$
Other units are also possible, such as $\text{cm/s}$ or $\text{km/h}$, but in physics, $\text{m/s}$ is standard.
Physical interpretation
Instantaneous velocity answers the question, "If I look at the object right now, how is its position changing?" It is a local description of motion, not an average over a long interval.
This is why a speedometer in a car is closer to instantaneous speed than average speed. It tries to tell you how fast the car is moving at that moment.
In one-dimensional motion, instantaneous velocity gives very detailed information. It tells whether the object is moving forward, backward, or not moving at that instant.
Key idea to remember
Average velocity uses a whole interval of time. Instantaneous velocity uses one exact instant and is found from the derivative of position.
The central formula for instantaneous velocity is
$$
v = \frac{dx}{dt}
$$
In words, instantaneous velocity is the time rate of change of position.
If you know the position function $x(t)$, differentiate it to get velocity. If you have a position-time graph, find the slope of the tangent line. These are the two main ways to understand instantaneous velocity.