Average Velocity and Instantaneous Velocity by David Halliday, Robert Resnick

If a particle moves from one point to another, we might need to know how fast it moves. We can define two quantities that deal with "how fast": average velocity and instantaneous velocity. However, here we must consider these quantities as vectors and use vector notation.

If a particle moves through a displacement $\overrightarrow{r}$ in a time interval Δ t, then its average velocity ${\overrightarrow{v}}_{\mathrm{avg}}$ is

$\text{average velocity}=\frac{\text{displacement}}{\text{time interval}}$

or ${\overrightarrow{v}}_{\mathrm{avg}}=\frac{\Delta \overrightarrow{v}}{\Delta t}$ (4-8)

This tells us that the direction of ${\overrightarrow{v}}_{\mathrm{avg}}$ (the vector on the left side of Eq. 4-8) must be the same as that of the displacement $\Delta \overrightarrow{r}$ (the vector on the right side). Using Eq. 4-4, we can write Eq. 4-8 in vector components as

${\overrightarrow{v}}_{\mathrm{avg}}=\frac{\Delta x\hat{i}+\Delta y\hat{j}+\Delta z\hat{k}}{\Delta t}=\frac{\Delta x}{\Delta t}\hat{i}+\frac{\Delta y}{\Delta t}\hat{j}+\frac{\Delta z}{\Delta t}\hat{k}$
(4-9)

For example, if a particle moves through displacement (12 m)$\hat{i}$+ (3.0 m)$\hat{k}$ in 2.0 s, then its average velocity during that move is

${\overrightarrow{v}}_{\mathrm{avg}}=\frac{\Delta \overrightarrow{v}}{\Delta t}=\frac{\left(12m\right)\hat{i}+\left(3.0m\right)\hat{k}}{2.0s}=(6.0m/s)\hat{i}+(1.5m/s)\hat{k}$

That is, the average velocity (a vector quantity) has a component of 6.0 m/s along the x axis and a component of 1.5 m/s along the z axis.

When we speak of the velocity of a particle, we usually mean the particle's instantaneous velocity $\overrightarrow{v}$ at some instant. This $\overrightarrow{v}$ is the value that ${\overrightarrow{v}}_{\mathrm{avg}}$ approaches in the limit as we shrink the time interval Δt to 0 about that instant. Using the language of calculus, we may write $\overrightarrow{v}$ as the derivative

${\overrightarrow{v}}_{}=\frac{d\overrightarrow{r}}{dt}$ (4-10)

Figure 4-3 The displacement $\Delta \overrightarrow{r}$ of a particle during a time interval $\Delta \stackrel{}{t}$, from position 1 with position vector ${\overrightarrow{r}}_1$ at time t₁ to position 2 with position vector ${\overrightarrow{r}}_2$ at time t₂. The tangent to the particle's path at position 1 is shown.

Figure 4-3 shows the path of a particle that is restricted to the xy plane. As the particle travels to the right along the curve, its position vector sweeps to the right. During time interval Δt, the position vector changes from ${\overrightarrow{r}}_1\text{ to }{\overrightarrow{r}}_2$ and the particle's displacement is Δr.

To find the instantaneous velocity of the particle at, say, instant t₁ (when the particle is at position 1), we shrink interval Δt to 0 about t₁. Three things happen as we do so. (1) Position vector ${\overrightarrow{r}}_2$ in Fig. 4-3 moves toward ${\overrightarrow{r}}_1$ so that Δr shrinks toward zero. (2) The direction of $\frac{\Delta \overrightarrow{r}}{\Delta t}$ (and thus of ${\overrightarrow{v}}_{\mathrm{avg}}$ ) approaches the direction of the line tangent to the particle's path at position 1. (3) The average velocity ${\overrightarrow{v}}_{\mathrm{avg}}$ approaches the instantaneous velocity $\overrightarrow{v}$ at t_1.

In the limit as $\Delta t\to 0$, we have ${\overrightarrow{v}}_{\mathrm{avg}}$ and, most important here, takes on the direction of the tangent line.Thus, $\overrightarrow{v}$ has that direction as well:

The direction of the instantaneous velocity $\overrightarrow{v}$ of a particle is always tangent to the particle's path at the particle's position.

The result is the same in three dimensions: $\overrightarrow{v}$ is always tangent to the particle's path. To write Eq. 4-10 in unit-vector form, we substitute for $\overrightarrow{r}$ from Eq. 4-1:

$\overrightarrow{v}=\frac{d}{dt}(x\hat{i}+y\hat{j}+z\hat{k})=\frac{dx}{dt}\hat{i}+\frac{dy}{dt}\hat{j}+\frac{dz}{dt}\hat{k}$

This equation can be simplified somewhat by writing it as

$\overrightarrow{v}=v_x\hat{i}+v_y\hat{j}+v_z\hat{k}$ (4-11)

where the scalar components of $\overrightarrow{v}$ are

$V_x=\frac{dx}{dt}\text{, }{V}_y=\frac{dy}{dt}\text{, and }{V}_z=\frac{dz}{dt}$ (4-12)

For example, dx/dt is the scalar component of $\overrightarrow{v}$ along the x axis. Thus, we can find the scalar components of $\overrightarrow{v}$ by differentiating the scalar components of $\overrightarrow{r}$.

Figure 4-4 The velocity $\overrightarrow{v}$ of a particle, along with the scalar components of $\overrightarrow{v}$

Figure 4-4 shows a velocity vector $\overrightarrow{v}$ and its scalar x and y components. Note that $\overrightarrow{v}$ is tangent to the particle's path at the particle's position. Caution: When a position vector is drawn, as in Figs4-3, it is an arrow that extends from one point (a "here") to another point (a "there"). However, when a velocity vector is drawn, as in Fig. 4-4, it does not extend from one point to another. Rather, it shows the instantaneous direction of travel of a particle at the tail, and its length (representing the velocity magnitude) can be drawn to any scale.

About the Authors

David Halliday was an American physicist known for his physics textbooks, Physics and Fundamentals of Physics, which he wrote with Robert Resnick. Both textbooks have been in continuous use since 1960 and are available in more than 47 languages.

Robert Resnick was a physics educator and author of physics textbooks. He was born in Baltimore, Maryland on January 11, 1923 and graduated from the Baltimore City College high school in 1939. He received his B.A. in 1943 and his Ph.D. in 1949, both in physics from Johns Hopkins University.

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