Physics Bootcamp

Subsection 4.3.1 Average Velocity

The ratio of displacement \(\Delta \vec r\) to the interval \(\Delta t\) over which displacement occured is called average velocity during that interval.

Since \(\Delta \vec r \) is a vector and we are dividing it by a simple number, average velocity, \(\vec v_{\text{av}} \) will also be a vector. Just as other vectors, we can represent velocity by an arrow of appropriate size and direction or analytically as components or in magnitude-direction form.

Noting that displacement \(\Delta \vec r \) has the components \((\Delta x, \Delta y, \Delta z) \text{,}\) we can also cast Eq. (4.3.1) in the component form to obtain the following for the components of the average velocity.

From the components, we can deduce the magnitude and direction as we would do for any other vector as illustrated in Checkpoint 4.3.1.

Subsection 4.3.2 Average Speed

Average speed is different from average velociy, although we often tend to confuse them as same. For average speed, we divide the total distance traveled on the actual path by time taken, not just the direct distance between the initial point and the final point.

The confusion comes from the fact that we use the same symbol for speed as we do for the magnitude of average velocity. To avoid the confusion, let us use symbol \(v_{s,av} \) for the average speed, \(d \) the distance traveled, and \(t \) the time, then we have

Average Speed is Different from Average Velocity: An extreme example that shows the difference between average speed and average velocity occurs in a cloed-loop motion. Let \(T\) denote the time it takes to return after going around the closed path, say from your home to the grocery store and back to your home. Let \(D\) be the total distance to the store and back. Average velocity here will be zero, since displacement is zero, but average speed will be \(D/T\text{.}\)

Subsection 4.3.3 Instantaneous Velocity - Formal Definition

Average velocity and average speed do not tell us motion moment-by-moment. For that Isaac Newton, introduced the concept of instantaneous velocity based on concepts of limit and infinitesimal intervals.

If you find average velocity using Eq. (4.3.1) for different interval values, i.e., different values of \(\Delta t\text{,}\) you will find that you get different velocities. Newton noticed that, as \(\Delta t\) become increasingly small, the differences in average velocities become less and less in a well-defined way such that it is possible to define a unique velocity in the limit \(\Delta t\rightarrow 0\text{.}\) Denoting the instantaneous velocity vector by \(\vec v \) and the position vector as \(\vec r\) we can write the definition of instantaneous velocity formally by

The magnitude of instantaneous velocity gives us the speed at that instant. Although, we noted above that average speed and magnitude of average velocity may be different, but instantaneous speed and magnitude of instantaneouos velocity are always same since averaging in Eq. (4.3.4) is over an infintesimal time interval.

Eq. (4.3.4) is an abstract definition, which we can make more accessibale by looking at the components. Denoting \(\vec r\left(\text{at } (t+\Delta t)\right) - \vec r\left(\text{at } t\right)\) by \(\Delta \vec r \equiv (\Delta x, \Delta y, \Delta z)\text{.}\)

These components of velocity are also called \(x\)-, \(y\)-, and \(z\)-velocity respectively. In a previous chapter, we have studied one-dimensional motion, where only one of them was non-zero. For general motion, you can expect all of them to be nonzero and you would need to keep track of them all.

Numbers \(v_x\text{,}\) \(v_y\text{,}\) and \(v_z\) you would obtain this way are the Cartesian components of the vector \(\vec v\text{.}\) Thus, once we have obtained \((v_x, v_y, v_z)\text{,}\) we can deduce the magnitude and direction of the velocity vector as we do for any vector. For instane, magnitude of velocity, i.e., speed, will simply be the magnitude of the vector.

For direction of \(\vec v\text{,}\) we will use either polar angle in case of a motion in a plane, or spherical cordinate angles in case of motion in a three-dimensional space.

Subsection 4.3.4 Velocity from Slopes of \((x, y, z) \) vs \(t \) Plots.

When we studied one dimensional motion on \(x\) axis, we noted that \(v_x\) was the derivative of \(x\) with respect to \(t\text{.}\) The value of derivative at a point is equal to the slope of the tangent to the function at that point. Therefore, we were able to compute \(v_x\) by a plot of \(x\) versus \(t\text{.}\)

More generally, velocity (vector) is derivative \(\vec r\) as defined in Eq. (4.3.4). We cannot plot \(\vec r\) versus \(t\) like a function, since a vector has both quantitative (the magnitude) and qualitative properties (the direction). But, we can think of \(\vec r\) as a composite of three functions, \(x(t)\text{,}\) \(y(t)\text{,}\) and \(z(t)\text{,}\) where I have placed \(t\) in the argument to emphasize the independent variable being \(t\) here. Plots of \(x(t)\text{,}\) \(y(t)\text{,}\) and \(z(t)\) versus \(t\) allows us to find \(v_x\text{,}\) \(v_y\text{,}\) and \(v_z\text{,}\) respectively.

Once we have obtained \((v_x,\ v_y,\ v_z)\text{,}\) we can deduce the magnitude and direction of the velocity vector as usual.

Subsection 4.3.5 Direction of Instantaneous Velocity from Trajectory

If you can draw the path of the moving object in space, you will get some curvy trajectory in space. The direction of velocity at some instant, then is the direction of the tangent to this trajectory as shown in Figure 4.3.10.

The trajectory of a projectile is shown in the figure below with the assumption that the initial velocity was \(80\text{ m/s}\) at the angle of \(72^{\circ}\) from the horizontal direction. The position of the projectile at successive \(1 \text{ sec}\) instants are shown in the figure

You can see that the direction as well as the magnitude of the velocity of the projectile changes with time. As the projectile rises, the distance covered in each subsequent second decreases as a result of slowing down. At the top of the trajectory, the projectile only has horizontal velocity since the vertical component of the velocity is zero there. The velocity is never pointed vertically straight down since the projectile always has a non-zero horizontal component of the velocity.

Subsection 4.3.6 (Calculus) Instantaneous Velocity and Instantanous Speed

To study accelerating motion, we need to understand not just average velocity, but also velocity throughout the motion. The instantaneous velocity and instantanous speed are mathematical concepts that tell us how fast the object is moving at an instant. Additionally, the instantaneous velocity also tells us which way the object is headed next.

We need Calculus to define these quantities - so if you have not had Calculus yet, you can skip this section - just note that the magnitude of the velocity at an instant says how fast it is moving at that instant, which is same as speed at that instant, and the direction of the velocity vector tells us the direction in which the object is headed next.

Velocity at an instant, called the instantaneous velocity or simply as velocity, is defined as the average velocity over infinitesimal interval around the instant of interest. This requires taking a limit of the average velocity as we decrease the infinitesimal inerval. This results in the velocity equal to the derivative of position vector \(\vec r\text{.}\)

One way to think about derivative of a vector is to write it in Cartesian components. Since \(\vec r = x \hat i + y \hat j + z \hat k\text{,}\) the Cartesian components of velocity vector can be obtained by taking derivatives of position coordinates.

Let us denote, as usual, the Catesian components of \(\vec v\) by \(v_x \text{,}\) \(v_y \text{,}\) and \(v_z \) . Then, we will have each component of the velocity vector equal to just a derivative of the corresponding coodinate.

Magnitude of Instantaneous Velocity. The magnitude of the velocity vector tells us how fast the object is moving. This is what we also mean by speed. Therefore, the magnitude of instantaneous velocity vector is the same thing as instantaneous speed. From the Cartesian components of the velocity, we can immediatrely write a formula for speed in terms of the derivatives of the coordinates.

Direction of Instantaneous Velocity. The above procedure provides a way of finding the components of the instantaneous velocity,, viz., \(\vec v = (v_x, v_y, v_z) \text{.}\) From components \((v_x, v_y, v_z) \text{,}\) we can also get the direction of \(\vec v\) in space as we would for any vector quantity. For instance, if the velocity vector was in the \(xy \)-plane, the \(z \) component will be zero, and we would write \(\vec v = (v_x, v_y) \text{.}\) Then, the angle with respect to the \(x \) axis will be

In case of a fully three-dimensional situation, we would use the polar and azimuthal angles, as usual, to indicate the direction.