Motion in a straight line

  • Motion is change in position of an object with time.
  • The motion of objects in a straight path is known as rectilinear motion.
  • In Kinematics, we study ways to describe motion without going into the causes of motion.

Distance and displacement

  • Path length is defined as the actual path traversed by body during motion in a given time interval of time.
  • Distance is a scalar quantity.
  • Displacement of a body in a given time is defined as the shortest distance between initial and final position of the object.
  • Displacement is a vector quantity.

Uniform motion

  • If an object moving along the straight line covers equal distances in equal intervals of time, it is said to be in uniform motion along a straight line.

Average velocity

  • Average velocity is defined as the change in position or displacement (Δx) divided by the time intervals (Δt), in which the displacement occurs:
\[\overline v = \frac{{{x_2} – {x_1}}}{{{t_2} – {t_1}}} = \frac{{\Delta x}}{{\Delta t}}\]

Where x1 and x2 are the positions of the object at time t1 and t2 respectively.

  • The SI unit for average velocity is m/s or ms-1.
  • Average velocity is a vector quantity.
  • The average velocity can be positive or negative depending upon the sign of the displacement.

Average speed

  • Average speed is defined as the total path length travelled divided by the total time interval during which the motion has taken place:
\[{\text{Average speed = }}\frac{{{\text{Total path length}}}}{{{\text{Total }}{\text{time interval}}}}\]
  • The SI unit for average speed is m/s or ms-1.
  • Average speed is a scalar quantity.
  • The average speed is always positive.

Instantaneous velocity

  • The velocity at an instant is defined as the limit of the average velocity as the time interval ∆t becomes infinitesimally small :
\[\mathop {\lim }\limits_{\Delta t \to 0} \frac{{\Delta x}}{{\Delta t}}\]
  • It is the rate of change of position with respect to time, at that instant.

Instantaneous speed

  • Instantaneous speed is the magnitude of the instantaneous velocity.

Acceleration

  • It is the rate of change of velocity with time.
  • acceleration can also be positive, negative or zero.

Average acceleration

  • The average acceleration ‘ā’ over a time interval is defined as the change of velocity divided by time interval:
\[\bar a = \frac{{{v_2} – {v_1}}}{{{t_2} – {t_1}}} = \frac{{\Delta v}}{{\Delta t}}\]

Where v1 and v2 are the Instantaneous velocities at time t1 and t2 respectively.

  • It is the average change of velocity per unit time.
  • The SI unit of acceleration is ms-2.

Instantaneous acceleration

  • It is defined as the acceleration of a body at particular instant:
\[{a_{inst}} = \mathop {\lim }\limits_{\Delta t \to 0} \frac{{\Delta v}}{{\Delta t}} = \frac{{dv}}{{dt}}\]
  • The acceleration at an instant is the slope of the tangent to the v-t curve at that instant.

Kinematic equations for uniformly accelerated motion

\[\begin{gathered} v{\text{ }} = {\text{ }}u{\text{ }} + {\text{ }}at{\text{ }} \hfill \\ s{\text{ }} = {\text{ }}ut{\text{ }} + {\text{ }}\frac{1}{2}a{t^2}{\text{ }} \hfill \\ {v^2} = {\text{ }}{u^2} + {\text{ }}2as \hfill \\ \end{gathered} \]

Relative velocity

  • The relative velocity is defined as the velocity of an object with respect to another observer.
  • The velocity of object B relative to object A is :
\[{v_{BA}} = {\text{ }}{v_B}-{\text{ }}{v_A}\]
  • The velocity of object A relative to object B is :
\[{v_{AB}} = {\text{ }}{v_A}-{\text{ }}{v_B}\]