An object is said to be in uniform motion, if it covers equal displacement in equal interval of time however, small these time intervals maybe.
The study of the uniform motion is the simplest one. And accelerated motion requires a cause or an effort. Uniform motion in a straight line is physically important from the point of view that it does not require any such cause or effort.
Consider that an object is in uniform motion along a straight line OX. Let point O be the origin for position measurement. Suppose that the origin for measurement of time is also taken at the instant, when the object is at point O.
Further consider that at any time t the object is at point A and at time t’, it reaches point B such that displacement $ \displaystyle \overrightarrow{{OA}}=\overrightarrow{x}$ and $ \displaystyle \overrightarrow{{OB}}=\overrightarrow{{{x}’}}$. Obviously, the displacement of the object in the time interval t and t’ i.e. t’ – t is $ \displaystyle \overrightarrow{{AB}}=\overrightarrow{{{x}’}}-\overrightarrow{x}$.
Now the velocity of the uniform motion $ \displaystyle \text{= }\frac{{\text{displacement}}}{{\text{time interval}}}$
$ \displaystyle \therefore \overrightarrow{v}=\dfrac{{\overrightarrow{{{x}’}}-\overrightarrow{x}}}{{\overrightarrow{{{t}’}}-\overrightarrow{t}}}$
The following points are true for uniform motion:
- Generally, the displacement may or may not be equal to the actual distance covered by an object. However, when the uniform motion takes place along a straight line in a given direction, the magnitude of the displacement is equal to the actual distance covered by the object.
- The velocity of uniform motion is same for different choices of t and t’.
- The velocity of uniform motion is not affected due to the shift of origin.
- The positive value of velocity means the object is moving towards right of the origin while the negative velocity means the motion is towards the left of the origin.
- As said earlier, for an object to be in uniform motion no cause or effort i.e no forces required.
- The average and instantaneous velocity in a uniform motion or always equal, as the velocity during uniform motion is same at each point of the path or at each instant.
Formula for Uniform Motion
When an object is in uniform motion, its velocity always remains constant. Mathematically, for uniform motion;
$ \displaystyle \overrightarrow{v}=\text{ constant vector }…\text{ (1)}$
Suppose that the origin of the position access is point O and the origin for the time measurement is taken as the instant, when the object is at point A, such that $ \displaystyle \overrightarrow{{OA}}=\overrightarrow{{{{x}_{\circ }}}}$. If at time t, the object moving with constant velocity $ \displaystyle \overrightarrow{v}$ is at point B, such that $ \displaystyle \overrightarrow{{OB}}=\overrightarrow{x}$, then
$ \displaystyle \overrightarrow{x}=\overrightarrow{{{{x}_{\circ }}}}+\overrightarrow{v}t\text{ }…\text{ (2)}$
Similarly, if at time t’, the object reaches point C, such that $ \displaystyle \overrightarrow{{OC}}=\overrightarrow{{{x}’}}$,
Then,
$ \displaystyle \overrightarrow{{{x}’}}=\overrightarrow{{{{x}_{\circ }}}}+\overrightarrow{v}{t}’\text{ }…\text{ (3)}$
Subtracting equation 2 from 3, we have
$ \displaystyle \begin{array}{l}\overrightarrow{{{x}’}}-\overrightarrow{x}=\left( {\overrightarrow{{{{x}_{\circ }}}}+\overrightarrow{v}{t}’} \right)-\left( {\overrightarrow{{{{x}_{\circ }}}}+\overrightarrow{v}t} \right)\\\Rightarrow \overrightarrow{{{x}’}}=\overrightarrow{x}+\overrightarrow{v}({t}’-t)\text{ }…\text{(4)}\end{array}$
The equations 2, 3 and 5 represent the kinematics of uniform motion along a straight line. The equation 4 may be obtained directly by expanding equation $ \displaystyle \overrightarrow{v}=\frac{{\overrightarrow{{{x}’}}-\overrightarrow{x}}}{{\overrightarrow{{{t}’}}-\overrightarrow{t}}}$ for $ \displaystyle {\overrightarrow{{{x}’}}}$ or the equation 4 may be used to obtain expression for velocity of the uniform motion.
If we call $ \displaystyle \overrightarrow{x}-\overrightarrow{{{{x}_{\circ }}}}=\overrightarrow{S}$ the displacement of the object in time t, then equation 2 becomes
$ \displaystyle \overrightarrow{S}=\overrightarrow{v}t$
In the for going study of motion along a straight line, the various displacements of the object such as $ \displaystyle \overrightarrow{{{{x}_{\circ }}}},\overrightarrow{x}\text{ or }\overrightarrow{{{x}’}}$ represent vectors having the same direction and consequently velocity vector is also in the same direction. Just for the sake of convenience, the arrowheads over the displacement and the velocity vectors may be dropped in the equations from 1 to 5.
