Motion in One Dimension (Algebra Based)
Table of contents
1. Distance & Displacement
An object is said to be in motion or at rest only when its position changes or remains the same with respect to a reference point. Consider point $P$ on a one-dimensional line (Fig 1). With the point being by itself, it is impossible to say anything about its position. It is also impossible to say whether the point is in motion or at rest.
To describe the position of point $P$, we need to introduce a reference point, say point $O$. When point $P$ changes its position relative to point $O$, it is said to be in motion. To describe this change quantitatively, we introduce two related but distinct concepts - distance and displacement.
Distance is the total length of the path traveled by an object, regardless of direction. it is a scalar quantity, meaning it only has magnitude and no direction.
Displacement, on the other hand, is the shortest straight-line path between the initial and final positions of an object. It is a vector quantity, which means it has both magnitude and direction.
The S.I. unit for both distance and displacement is the meter ($m$).
For example, overlay a coordinate system on the line such that point $O$ is at the origin (Fig 3). Suppose point $P$ initially at the origin moves $x_1$ units to the right of point $O$ and then moves to a new position $x_2$ units to the left of point $O$. In this case, the distance traveled by point $P$ is $x_1 + x_2$ units. However, the displacement of point $P$ is given by the difference between its final and initial positions, which is $-x_2 - 0 = -x_2$ units (negative sign indicates direction to the left).
Example 1: Suppose a particle moves from $x=0m$ to $x=5m$.
- Distance traveled: $5m$.
- Displacement: final position - initial position $=5m - 0m$ $= 5m$.
Example 2: Suppose a car moves from $x_0=0m$ to $x_1=8m$, then reverses the direction and moves to $x_2=-5m$.
- Distance traveled: $|x_1 - x_0| + |x_2 - x_1|=$ $|8m - 0m| + |-5m - 8m| =$ $8m + 13m = 21m$.
- Displacement: final position - initial position $= x_2 - x_0 =$ $-5m - 0m = -5m$.
Example 3: Suppose the car in Example 2 was initially at $x_0 = 3m$ instead of $x_0 = 0m$.
- Distance traveled: $|x_1 - x_0| + |x_2 - x_1|=$ $|8m - 3m| + |-5m - 8m| =$ $5m + 13m = 18m$.
- Displacement: final position - initial position $= x_2 - x_0 =$ $-5m - 3m = -8m$.
Therefore, in mathematical terms, the displacement $\Delta x$ of an object moving from an initial position $x_i$ to a final position $x_f$ is given by: $$\Delta x = x_f - x_i$$
2. Speed & Velocity
In the previous section, we learned that distance and displacement describe how far an object moves. To describe how fast the motion occurs, we introduce speed and velocity.
Speed is the rate at which an object covers distance. It tells us how fast an object is moving, regardless of its direction. Mathematically, speed $s$ is defined as the total distance $d$ traveled divided by the time $t$ taken to travel that distance: $$s = \frac{d}{t}$$ Since distance is a scalar quantity, and distance traveled is always positive, speed is also a scalar quantity and is always non-negative.
Velocity is the rate of change of displacement with respect to time. It tells us how fast an object is moving in a specific direction. For instance, if by convention, we consider rightward motion as positive and leftward motion as negative, then velocity can be positive or negative based on the direction of motion. Mathematically, velocity $v$ is defined as the displacement $\Delta x$ divided by the time interval $\Delta t$ during which the displacement occurs: $$v = \frac{\Delta x}{\Delta t}$$ Because displacement is a vector quantity, velocity is also a vector quantity. Therefore, at any given instant, the velocity of an object represents both its speed and the direction of its motion.
S.I. unit of Speed & Velocity
The S.I. unit for time is the second ($s$). And from chapter 1, we know that the S.I. unit for both distance and displacement is the meter ($m$). To obtain the S.I. unit for speed and velocity, we can substitue these units into the above definitions. Therefore, the S.I. unit for speed is, $$[s] = \frac{[d]}{[t]} = \frac{m}{s}$$ Similarly, the S.I. unit for velocity is $m/s$.