Physics Bootcamp

Subsection 4.4.1 Average Acceleration

Suppose we are looking at what happens to velocity during an interval from \(t_i \) to \(t_ f \text{.}\) Let the instantaneous velocity (vector) at \(t_i \) be \(\vec v_i \) and at \(t_f \) be \(\vec v_f \text{.}\) Then, average acceleration is defined by

Writing the change in velocity as \(\Delta v\) and intertval of time as \(\Delta t\text{,}\) we write this definition more succinctly as

Since we are dividing a vector \(\Delta \vec v\) by a number \(\Delta t\text{,}\) average acceleration is also a vector. The direction of \(\vec a_{\text{av}}\) will be in the direction of change in velocity \(\Delta \vec v\) and the magnitude will the magnitude of vector \(\Delta \vec v\) divided by the interval of time. In particular, the direction of acceleration has nothing to do with direction of velocity at any one instant, but instead with the change in velocity, i.e, how velocity of changing.

Figure 4.4.1 shows an object moving in a circle with constant speed of \(5\text{ m/s}\text{.}\) In (a), we see that the change in velocity between A and B points of the circle are neither in the direction of \(\vec v_A\) nor in the direction of \(\vec v_B\text{,}\) but its own direction as shown by vector \((\vec v_B - \vec v_A)\text{.}\) Since the change in velocity is not zero, the object's average acceleration between A and B is not zero. In (b), another construction for change in velocity during B and C is shown. Try to draw for what happens between C and D and betwen D and A.

Just as with other vector quantities, we can analyze average acceleration in terms of its Cartesian components, \(a_{av,x} \text{,}\) \(a_{av,y} \text{,}\) and \(a_{av,z} \text{.}\) From these components, you can find the magnitude and direction of the acceleration vector.

Subsection 4.4.2 Instantaneous Acceleration - Formal Definition

Average acceleration defined above does not tell us about moment-by-moment acceleration. For that Isaac Newton, introduced the concept of instantaneous acceleration, or more siply acceleration, based on concepts of limit and infinitesimal intervals, as we have done for instantaneous velocity.

If you find average acceleration using Eq. (4.4.2) for different interval values, i.e., different values of \(\Delta t\text{,}\) you will find that you get different velocities as was illustrated for a circular motion. Newton noticed that, when \(\Delta t\) becomes increasingly small, the differences in average acceleration so-obtained become less and less in a well-defined way such that it is possible to define a unique acceleration in the limit \(\Delta t\rightarrow 0\text{.}\) Denoting the instantaneous acceleration vector by \(\vec a \) and instantaneous velocity vector as \(\vec v\) we can write the definition of instantaneous velocity formally by

Unlike the magnitude of instantaneous velocity, which is called speed, the magnitude of instantaneous acceleration does not have any special name.

Eq. (4.4.3) is an abstract definition, which we can make more accessibale by looking at the components. Denoting \(\vec v\left(\text{at } (t+\Delta t)\right) - \vec v\left(\text{at } t\right)\) by \(\Delta \vec v \equiv (\Delta v_x, \Delta v_y, \Delta v_z)\text{.}\)

These components of acceleration are also called \(x\)-, \(y\)-, and \(z\)-accelerations 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.

Subsection 4.4.3 Acceleration from Slopes of \((v_x,\ v_y,\ v_z) \) vs \(t \) Plots.

Acceleration at an instant, also called the instantaneous acceleration, is the rate at which velocity is changing with time. In one-dimensional situation, we analyzed by only \(x\) component of acceleration for motion along \(x\)-axis. Analytically, this was given by the derivative of \(v_x\text{,}\) the \(x\)-component of velocity.

Since derivate of a function also equals slope of a graph of that function, we had used \(v_x\) versus \(t\) plot to find \(a_x\) when using graphical method.

Now, with acceleration possible along arbitrary directions, we can choose to analyze using all three Cartesian directions and calculate derivatives from the slopes of appropriate plots. Thus,

Once we have obtained \((a_x,\ a_y,\ a_z)\text{,}\) we can deduce the magnitude and direction of the acceleration vector as usual. For instance, magnitude of acceleration will be

and the direction will be given by one angle if the motion is 2-dimensional or by two angles if the motion is 3-dimensional.

Subsection 4.4.4 (Calculus) Instantaneous Acceleration

To obtain acceleration at a particular instant \(t \text{,}\) we seek average acceleration in an infinitesimal interval near the instant \(t \text{.}\) This leads to the ratio in Eq. (4.4.2) becoming a derivative.

We can express the Cartesian components of acceleration as derivatives of the corresponding Cartesian components of the velocity vector.

Since \(v_x = dx/dt \text{,}\) \(v_y = dy/dt \text{,}\) snd \(v_z = dz/dt \text{,}\) we find that we can express components of acceleration as second derivatives of the position.

This is often written compactly as