Physics · Chapter 3
Study notes aligned to the official NEB syllabus.
Mechanics is the branch of physics that deals with the state of rest or motion of bodies under the action of forces. It is broadly divided into three parts:
This chapter deals with kinematics in a straight line and in two dimensions (projectile motion). We describe motion using the quantities displacement, velocity, and acceleration, represent it with graphs, derive the equations of uniformly accelerated motion, apply them to motion under gravity, and finally analyse projectile motion.
Worked question. Can a body have zero displacement but a non-zero distance travelled?
Yes. If a body starts from a point, moves along a path, and returns to its starting point, its displacement is zero (initial and final positions coincide) while the distance travelled (the total path length) is non-zero. For example, an athlete completing one full lap of a circular track has zero displacement but a distance equal to the circumference.
$$\text{Speed} = \frac{\text{distance travelled}}{\text{time taken}}$$
It is a scalar quantity and its SI unit is $\text{m s}^{-1}$.
$$\text{Velocity} = \frac{\text{displacement}}{\text{time taken}}$$
It is a vector quantity (directed along the displacement) and its SI unit is $\text{m s}^{-1}$.
$$\text{Average speed} = \frac{\text{total distance travelled}}{\text{total time taken}}$$
$$ \begin{aligned} \vec{v}_{\text{av}} &= \frac{\text{total displacement}}{\text{total time taken}} \ &= \frac{\Delta \vec{x}}{\Delta t} \end{aligned} $$
Instantaneous speed is the speed of a body at a particular instant of time on its path. It is the magnitude of the instantaneous velocity.
Instantaneous velocity is the velocity of a body at a particular instant of time. It is obtained by making the time interval infinitesimally small:
$$ \begin{aligned} \vec{v} &= \lim_{\Delta t \to 0} \frac{\Delta \vec{x}}{\Delta t} \ &= \frac{d\vec{x}}{dt} \end{aligned} $$
Geometrically, the instantaneous velocity is the slope (gradient) of the displacement-time graph at that instant.
$$ \begin{aligned} \vec{a} &= \frac{\Delta \vec{v}}{\Delta t} \ \qquad \vec{a} &= \lim_{\Delta t \to 0}\frac{\Delta \vec{v}}{\Delta t} \ &= \frac{d\vec{v}}{dt} \end{aligned} $$
It is a vector quantity and its SI unit is $\text{m s}^{-2}$ (dimensions $[\mathrm{M^0,L,T^{-2}}]$).
Example (constant acceleration from a relation). If the displacement of a body is proportional to the square of time, state the nature of the motion.
Let the displacement be $y = kt^2$, where $k$ is a constant. Then:
$$ \begin{aligned} v &= \frac{dy}{dt} \ &= \frac{d}{dt}(kt^2) \ &= 2kt \ a &= \frac{dv}{dt} \ &= \frac{d}{dt}(2kt) \ &= 2k \ &= \text{constant} \end{aligned} $$
Since the acceleration is constant, the body moves with uniform (constant) acceleration.
The relative velocity of a body A with respect to a body B is the velocity with which A appears to move as seen by an observer sitting on B. It is obtained by subtracting the velocity of B from the velocity of A (vector subtraction):
$$\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$$
Similarly, the velocity of B relative to A is $\vec{v}_{BA} = \vec{v}B - \vec{v}A$, so that $\vec{v}{AB} = -\vec{v}{BA}$; the two relative velocities are equal in magnitude but opposite in direction.
Special cases (motion along a straight line):
$$v_{AB} = v_A - v_B$$
Two cars moving the same way at nearly equal speeds appear almost at rest with respect to each other.
$$v_{AB} = v_A + v_B$$
This is why two trains passing in opposite directions seem to rush past very quickly.
General case (two dimensions): if A and B move at an angle $\theta$ to each other, the magnitude of the relative velocity is found from the vector triangle:
$$v_{AB} = \sqrt{v_A^{2} + v_B^{2} - 2 v_A v_B \cos\theta}$$
Relative velocity is useful in problems such as a boat crossing a flowing river, rain falling on a moving person, and two bodies moving under gravity (their relative acceleration is zero because both fall with the same $g$).
A displacement-time graph plots displacement on the $y$-axis against time on the $x$-axis.
$$ \begin{aligned} \text{velocity} &= \frac{\Delta s}{\Delta t} \ &= \text{slope of the } s\text{-}t \text{ graph} \end{aligned} $$
A velocity-time graph plots velocity on the $y$-axis against time on the $x$-axis.
$$ \begin{aligned} \text{acceleration} &= \frac{\Delta v}{\Delta t} \ &= \text{slope of the } v\text{-}t \text{ graph} \end{aligned} $$
$$\text{displacement} = \text{area under the } v\text{-}t \text{ graph}$$
These three geometric facts (slope of $s$-$t$ = velocity, slope of $v$-$t$ = acceleration, area under $v$-$t$ = displacement) are the basis for deriving the equations of motion below.
Mechanics is the branch of physics that deals with the state of rest or motion of bodies under the action of forces. It is broadly divided into three parts:
This chapter deals with kinematics in a straight line and in two dimensions (projectile motion). We describe motion using the quantities displacement, velocity, and acceleration, represent it with graphs, derive the equations of uniformly accelerated motion, apply them to motion under gravity, and finally analyse projectile motion.
Worked question. Can a body have zero displacement but a non-zero distance travelled?
Yes. If a body starts from a point, moves along a path, and returns to its starting point, its displacement is zero (initial and final positions coincide) while the distance travelled (the total path length) is non-zero. For example, an athlete completing one full lap of a circular track has zero displacement but a distance equal to the circumference.
It is a scalar quantity and its SI unit is .
It is a vector quantity (directed along the displacement) and its SI unit is .
Instantaneous speed is the speed of a body at a particular instant of time on its path. It is the magnitude of the instantaneous velocity.
Instantaneous velocity is the velocity of a body at a particular instant of time. It is obtained by making the time interval infinitesimally small:
Geometrically, the instantaneous velocity is the slope (gradient) of the displacement-time graph at that instant.
It is a vector quantity and its SI unit is (dimensions ).
Example (constant acceleration from a relation). If the displacement of a body is proportional to the square of time, state the nature of the motion.
Let the displacement be , where is a constant. Then:
Since the acceleration is constant, the body moves with uniform (constant) acceleration.
The relative velocity of a body A with respect to a body B is the velocity with which A appears to move as seen by an observer sitting on B. It is obtained by subtracting the velocity of B from the velocity of A (vector subtraction):
Similarly, the velocity of B relative to A is , so that ; the two relative velocities are equal in magnitude but opposite in direction.
Special cases (motion along a straight line):
Two cars moving the same way at nearly equal speeds appear almost at rest with respect to each other.
This is why two trains passing in opposite directions seem to rush past very quickly.
General case (two dimensions): if A and B move at an angle to each other, the magnitude of the relative velocity is found from the vector triangle:
Relative velocity is useful in problems such as a boat crossing a flowing river, rain falling on a moving person, and two bodies moving under gravity (their relative acceleration is zero because both fall with the same ).
A displacement-time graph plots displacement on the -axis against time on the -axis.
A velocity-time graph plots velocity on the -axis against time on the -axis.
These three geometric facts (slope of - = velocity, slope of - = acceleration, area under - = displacement) are the basis for deriving the equations of motion below.