### How the Kinematic Equations are Related to Acceleration

We are studying concepts related to motion: time, displacement, velocity, and especially acceleration. We are only concerned with motion in one dimension. The kinematic equations apply to conditions of constant acceleration and show how these concepts are related. Constant acceleration is acceleration that does not change over time. The first kinematic equation relates displacement *d*, average velocity $\overline{v}$, and time *t*.

3.4$$d={d}_{0}+\overline{v}t\phantom{\rule{0.25}{0ex}}\text{, the initial displacement}\phantom{\rule{0.25em}{0ex}}{d}_{0}\phantom{\rule{0.25}{0ex}}\text{is often 0, in which case the equation can be written as}\phantom{\rule{0.25}{0ex}}\overline{v}=\frac{d}{t}$$

This equation says that average velocity is displacement per unit time. We will express velocity in meters per second. If we graph displacement versus time, as in Figure 3.7, the slope will be the velocity. Whenever a rate, such as velocity, is represented graphically, time is usually taken to be the independent variable and is plotted along the *x* axis.

The second kinematic equation, another expression for average velocity $\overline{v},$
is simply the initial velocity plus the final velocity divided by two.

3.5$$\overline{v}=\frac{{v}_{0}+{v}_{\text{f}}}{2}$$

Now we come to our main focus of this chapter; namely, the kinematic equations that describe motion with constant acceleration. In the third kinematic equation, acceleration is the rate at which velocity increases, so velocity at any point equals initial velocity plus acceleration multiplied by time

3.6$$v={v}_{0}+at\phantom{\rule{0.2em}{0ex}}\text{Also},\text{if we start from rest (}{v}_{0}=0\text{), we can write}\phantom{\rule{0.2em}{0ex}}a=\frac{v}{t}$$

Note that this third kinematic equation does not have displacement in it. Therefore, if you do not know the displacement and are not trying to solve for a displacement, this equation might be a good one to use.

The third kinematic equation is also represented by the graph in Figure 3.8.

The fourth kinematic equation shows how displacement is related to acceleration

3.7$$d={d}_{0}+{v}_{0}t+\frac{1}{2}a{t}^{2}\text{.}$$

When starting at the origin, ${d}_{0}=0$ and, when starting from rest, ${v}_{0}=0$, in which case the equation can be written as

$$a=\frac{2d}{{t}^{2}}\text{.}$$

This equation tells us that, for constant acceleration, the slope of a plot of 2*d* versus *t*^{2} is acceleration, as shown in Figure 3.9.

The fifth kinematic equation relates velocity, acceleration, and displacement

3.8$${v}^{2}={v}_{0}^{2}+2a\left(d-{d}_{0}\right)\text{.}$$

This equation is useful for when we do not know, or do not need to know, the time.

When starting from rest, the fifth equation simplifies to

$$a=\frac{{v}^{2}}{2d}\text{.}$$

According to this equation, a graph of velocity squared versus twice the displacement will have a slope equal to acceleration.

Note that, in reality, knowns and unknowns will vary. Sometimes you will want to rearrange a kinematic equation so that the knowns are the values on the axes and the unknown is the slope. Sometimes the intercept will not be at the origin (0,0). This will happen when *v*_{0} or *d*_{0} is not zero. This will be the case when the object of interest is already in motion, or the motion begins at some point other than at the origin of the coordinate system.

### Virtual Physics

#### The Moving Man (Part 2)

Look at the Moving Man simulation again and this time use the *Charts* view. Again, vary the velocity and acceleration by sliding the red and green markers along the scales. Keeping the velocity marker near zero will make the effect of acceleration more obvious. Observe how the graphs of position, velocity, and acceleration vary with time. Note which are linear plots and which are not.

Grasp Check

On a velocity versus time plot, what does the slope represent?

- Acceleration
- Displacement
- Distance covered
- Instantaneous velocity

Grasp Check

On a position versus time plot, what does the slope represent?

- Acceleration
- Displacement
- Distance covered
- Instantaneous velocity

The kinematic equations are applicable when you have constant acceleration.

- $d={d}_{0}+\overline{v}t$, or $\overline{v}=\frac{d}{t}$ when
*d*_{0} = 0
- $\overline{v}=\frac{{v}_{0}+{v}_{\text{f}}}{2}$
- $v={v}_{0}+at$, or $a=\frac{v}{t}$ when
*v*_{0} = 0
- $d={d}_{0}+{v}_{0}t+\frac{1}{2}a{t}^{2}$, or $a=\frac{2d}{{t}^{2}}$ when
*d*_{0} = 0 and *v*_{0} = 0
- ${v}^{2}={v}_{0}^{2}+2a\left(d-{d}_{0}\right)$, or $a=\frac{2d}{{t}^{2}}$ when
*d*_{0} = 0 and *v*_{0} = 0