1.2 Motion in One Dimension

No doubt you will have noticed that the world is three dimensional. However it is certainly beneficial to study motion in a simplified setting before extending our ideas to the full 3D application. In Newtonian mechanics, we label each point in space with a continuous variable $x$ , and we define an origin where $x=0$ . This defines our coordinate system.

1.2.1 Position as a Function of Time

We can describe the motion of an object by writing its position $x$ as a function of a continuous parameter $t$ , which describes the passage of time from a reference point $t=0$ .

Definition 1.1.

The trajectory of an object is a function $x(t)$ where $t,x(t)\in\mathbb{R}$ . $x(t_{0})=x_{0}$ describes the position of the object $x_{0}$ at some time $t_{0}$ .

In classical mechanics, time evolves as the same rate for the whole universe. Our choice of coordinate system together with our choice of reference point for time is called a reference frame. By choosing our reference frame cleverly, we can often simplify problems. For example, if we were studying a block sliding down a slope, we could simplify the problem by rotating our coordinate system so that the $x$ axis lies parallel to the slope. In general, it is often best to align the $x$ axis with the direction of motion.

Definition 1.2.

The displacement of an object is the difference in positions between two times $t_{1}$ and $t_{2}>t_{1}$ . In one dimension:

\Delta x=x(t_{2})-x(t_{1})=x_{2}-x_{1}.

(1.1)

If $x(0)=x_{0}$ , then the total displacement of an object as a function of time is given by

s(t)=x(t)-x_{0}.

(1.2)

The distance travelled by an object between $t_{1}$ and $t_{2}$ is given by the magnitude of displacement,

\mathrm{distance}=\lvert\Delta x\rvert=\lvert x(t_{2})-x(t_{1})\rvert=\lvert x% _{2}-x_{1}\rvert.

(1.3)

In general, distance $\neq$ displacement. This is because displacement is a vector quantity, meaning it has direction and magnitude, whereas distance is a scalar quantity. In one dimension, the only difference between vector and scalar quantities is that vector quantities are signed. The distinction becomes greater in more dimensions when vector quantities are actually represented as vectors.

1.2.2 Derivatives of Position

Definition 1.3.

The instantaneous velocity, or just the velocity of an object is defined as the rate of change of the object’s position with respect to time, or simply the derivative.

v(t)=\lim_{\Delta t\to 0}\frac{x(t+\Delta t)-x(t)}{\Delta t}=\frac{% \operatorname{\mathrm{d}}\!x(t)}{\operatorname{\mathrm{d}}\!t}.

(1.4)

Velocity is also a vector quantity. The corresponding scalar quantity is speed, defined as the magnitude of velocity,

\mathrm{speed}=\lvert v(t)\rvert.

(1.5)

The average velocity of an object in a time interval $\Delta t=t_{2}-t_{1}$ is given by the displacement over the time interval, or also the time average of the velocity (which must be computed with an integral since it is a continuous property).

\bar{v}=\frac{\mathrm{total\ displacement}}{\mathrm{total\ time}}=\frac{\Delta x% }{\Delta t}=\frac{1}{\Delta t}\int_{t_{1}}^{t_{2}}v(t)\operatorname{\mathrm{d}% }\!t.

(1.6)

Meanwhile, the average speed is given by

\overline{\mathrm{speed}}=\frac{\mathrm{total\ distance}}{\mathrm{total\ time}% }=\frac{1}{\Delta t}\int_{t_{1}}^{t_{2}}\lvert v(t)\rvert\operatorname{\mathrm% {d}}\!t.

(1.7)

Here, we may identify

\Delta x=\int_{t_{1}}^{t_{2}}v(t)\mathrm{d}t,\quad\mathrm{total\ distance}=% \int_{t_{1}}^{t_{2}}\lvert v(t)\rvert\operatorname{\mathrm{d}}\!t.

(1.8)

If the object is travelling at constant velocity, then the integral is trivial and we recover (setting $t_{1}=0$ ) the equation that you probably learned in school,

\mathrm{distance}=\mathrm{speed}\times\mathrm{time}.

(1.9)

Definition 1.4.

The instantaneous acceleration, or just the acceleration of an object is defined as the rate of change of the object’s velocity with respect to time.

a(t)=\lim_{\Delta t\to 0}\frac{v(t+\Delta t)-v(t)}{\Delta t}=\frac{% \operatorname{\mathrm{d}}\!v(t)}{\operatorname{\mathrm{d}}\!t}=\frac{% \operatorname{\mathrm{d}}\!^{2}x(t)}{\operatorname{\mathrm{d}}\!t^{2}}.

(1.10)

Likewise, the average acceleration of an object in a time interval $\Delta t$ is

\bar{a}=\frac{\Delta v}{\Delta t}=\frac{1}{\Delta t}\int_{t_{1}}^{t_{2}}a(t)% \operatorname{\mathrm{d}}\!t.

(1.11)