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Mechanics

Motion in One, Two, and Three Dimensions

Motion is the continuous change in position of a body with respect to time and a given frame of reference. It is quantified using fundamental kinematic variables including displacement, velocity, and acceleration.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Uniform and uniformly accelerated motion in a straight line, equations of motion, relative velocity, projectiles under gravity, and graphical analysis.

PGT extension

General curvilinear motion in polar and Cartesian coordinates, vector treatment of kinematics, motion under central forces, and frames of reference including non-inertial systems.

Prerequisites

Vector algebra (dot product, cross product, components)
Calculus (differentiation and integration of functions)
Fundamental units and dimensions in SI

Kinematic Quantities and Definitions

Displacement is a vector quantity representing the shortest straight-line distance from the initial to the final position, with dimensions [L].
Average velocity is the ratio of total displacement to total time interval, whereas instantaneous velocity is the time derivative of the position vector r(t).
Acceleration measures the temporal rate of change of velocity vector, possessing components along the tangent and the normal to the trajectory.

Motion with Uniform Acceleration

Under constant acceleration a, the kinematic equations are valid and independent of the mass of the moving body.
The area under the velocity-time graph represents the total displacement of the particle over the given time interval.
Free fall near the Earth's surface represents vertical motion under a constant downward gravitational acceleration g approx 9.8 m/s².

Projectile Motion in Two Dimensions

Projectile motion under uniform gravity consists of independent horizontal motion with zero acceleration and vertical motion with constant downward acceleration g.
The trajectory of a projectile in a uniform gravitational field in the absence of air resistance is strictly parabolic.
The maximum horizontal range for a given initial launch speed is achieved at an angle of 45° to the horizontal.

Relative Motion and Frames of Reference

Galilean transformation relates the position coordinates of a particle between two inertial frames moving with a constant relative velocity.
In non-inertial frames accelerating at a, fictitious forces such as the pseudo-force ma and the Coriolis force must be introduced to apply Newton's laws.
The relative velocity of body A with respect to body B is given vectorially by v_AB = v_A - v_B.

Derivations

Equation of Parabolic Trajectory for a Projectile

Conditions: Uniform gravitational field, constant initial speed u, launch angle theta, zero air resistance

Resolve initial velocity into components: u_x = u cos(theta) and u_y = u sin(theta).
Express horizontal position as a function of time t: x = (u cos(theta))t, yielding $t = x / (u \cos(\theta))$.
Express vertical position: $y = (u \sin(\theta))t - (1/2)gt^{2}$.
Substitute the expression for time t into the vertical position equation to eliminate time.

Result: $y = x \tan(\theta) - \frac{gx^2}{2u^2 \cos^2(\theta)}$

Derivation of Maximum Height and Time of Flight

Conditions: Projectile launched from ground level with initial speed u at angle theta

At maximum height H, the vertical component of velocity becomes zero: v_y = u sin(theta) - gt_up = 0.
Solve for time to reach maximum height: t_up = (u sin(theta)) / g, and double it for total time of flight $T = (2u \sin(\theta)) / g$.
Substitute t_up into the vertical displacement equation $H = (u \sin(\theta))t_{up} - (1/2)gt_{up}^{2}$.

Result: $H = \frac{u^2 \sin^2(\theta)}{2g} \quad \text{and} \quad T = \frac{2u \sin(\theta)}{g}$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
First Equation of Motion	$$v = u + at$$	v = final velocity, u = initial velocity, a = acceleration, t = time m/s	Constant acceleration
Second Equation of Motion	$$s = ut + \frac{1}{2}at^2$$	s = displacement, u = initial velocity, a = acceleration, t = time m	Constant acceleration
Third Equation of Motion	$$v^2 = u^2 + 2as$$	v = final velocity, u = initial velocity, a = acceleration, s = displacement m²/s²	Constant acceleration
Horizontal Range of Projectile	$$R = \frac{u^2 \sin(2\theta)}{g}$$	R = range, u = initial speed, theta = launch angle, g = acceleration due to gravity m	Launched and landing on the same horizontal plane
Instantaneous Velocity Vector	$$v = \frac{dr}{dt}$$	v = velocity vector, r = position vector, t = time m/s	General motion
Tangential and Normal Acceleration	$$a = \frac{dv}{dt}\hat{t} + \frac{v^2}{R}\hat{n}$$	v = speed, R = radius of curvature, t_hat and $n_hat = unit vectors$ m/s²	Curvilinear motion in a plane
Relative Velocity	$$v_{AB} = v_A - v_B$$	$v_AB = velocity of A relative$ to B, $v_A = velocity of A$, $v_B = velocity of B$ m/s	Inertial frames
Coriolis Acceleration	$$a_c = -2(\omega \times v)$$	omega = angular velocity of rotating frame, v = velocity of particle in rotating frame m/s²	Rotating non-inertial frame

Graphs and experiments

Displacement-Time Graph

Draw: Position s on the vertical axis against time t on the horizontal axis for uniform, accelerated, and decelerated motion.

Interpret: The slope of the tangent at any point on the curve yields the instantaneous velocity at that precise instant.

Exam use: Frequently tested in TGT for identifying stationary bodies, constant velocity, and increasing acceleration phases.

Velocity-Time Graph

Draw: Velocity v on the vertical axis against time t on the horizontal axis.

Interpret: The slope of the line gives the acceleration, and the net area enclosed between the curve and the time axis represents the total displacement.

Exam use: Tested via multi-stage numerical problems requiring total distance and displacement calculations.

Projectile Trajectory Graph

Draw: Vertical height y versus horizontal distance x showing a symmetric parabolic arc.

Interpret: Demonstrates the independence of horizontal and vertical motions under uniform gravity.

Exam use: Used to evaluate maximum height, time of flight, and coordinates at arbitrary points along the path.

Solved numericals

TGT core

A stone is dropped from the top of a tower 125 m high. Calculate the time taken to reach the ground and its final velocity just before striking the ground. (Take $g = 10 m/s^{2})$

Identify given parameters: initial velocity u = 0, displacement s = 125 m, acceleration $a = g = 10 m/s^{2}$.
Use the kinematic equation $s = ut + (1/2)gt^{2}: 125 = 0 + (1/2)(10)t^{2}$.
Solve for t: 5t$^{2} = 125 => t^{2} = 25 => t = 5 s$.
Calculate final velocity using $v = u + gt: v = 0 + (10)(5) = 50 m/s$.

Answer: Time = 5 s, Final velocit$y = 50 m/s$

TGT core

A projectile is fired with an initial velocity of 20 m/s at an angle of 30 degrees to the horizontal. Calculate the maximum height reached. ($g = 10 m/s^{2})$

Identify parameters: $u = 20 m/s$, thet$a = 30°$, $g = 10 m/s^{2}$.
Recall the formula for maximum height: $H = (u^2 \sin^2(\theta)) / (2g)$.
Substitute values: sin(30°) = 0.5, so sin²(30°) = 0.25.
$H = (20^{2} * 0.25) / (2 * 10) = (400 * 0.25) / 20 = 100 / 20 = 5 m$.

Answer: 5 m

PGT extension

The position vector of a particle is given by r(t) = (3t^2)i + (2t^3)j + 5k in SI units. Find the magnitude of its acceleration at t = 2 s.

Find velocity vector by differentiating position with respect to time: v(t) = dr/d$t = (6t)i + (6t^2)j$.
Find acceleration vector by differentiating velocity with respect to time: a(t) = dv/d$t = (6)i + (12t)j$.
Substitute $t = 2 s into the acceleration vector: a(2) = 6i + (12 * 2)j = 6i + 24j$.
Calculate the magnitude of the acceleration vector: |a| = sqrt(6^2 + 24^2) = sqrt(36 + 576) = sqrt(612) = 6 sqrt(17) m/s².

Answer: 6sqrt(17) m/s²

Assumptions and limitations

Newtonian mechanics assumes classical spacetime where time is absolute and spatial measurements are independent of inertial reference frame speeds.
Air resistance is neglected in standard elementary projectile problems unless specified by a viscous drag force term.
The Earth is treated as an inertial frame for most elementary laboratory kinematics, ignoring Coriolis and centrifugal effects for short-duration local motions.

Common misconceptions

Common error: At the highest point of a projectile's trajectory, both velocity and acceleration are zero.

Correct: Only the vertical component of velocity is zero; horizontal velocity remains constant, and vertical acceleration is always equal to g downward.

Gravity acts continuously throughout the entire flight trajectory, meaning acceleration never vanishes.

Common error: Heavier objects fall faster than lighter objects in a vacuum.

Correct: All objects fall with identical acceleration under gravity regardless of their mass.

Galileo's principle and Newton's second law show that gravitational force (mg) divided by mass (m) yields an acceleration independent of mass.

Common error: Distance and displacement are always equal in magnitude.

Correct: Displacement is equal to distance only when motion occurs along a straight line in a single direction without turning back.

Distance is a scalar representing path length, whereas displacement is a vector connecting initial and final positions.

Quick revision

Kinematics describes motion without considering its causes, relying on position, velocity, and acceleration.
The slope of a velocity-time graph represents acceleration, and its area represents displacement.
Projectile motion combines uniform horizontal velocity with uniformly accelerated vertical motion.
Maximum projectile range on level ground occurs at a launch angle of 45 degrees.

Must remember

Displacement is a vector quantity representing the shortest straight-line distance from initial to final position with dimensions $[L]$.
The trajectory of a projectile in a uniform gravitational field without air resistance is strictly parabolic.
The maximum horizontal range for a given initial launch speed is achieved at an angle of $45^\circ$ to the horizontal.
In non-inertial frames accelerating at $a$, fictitious forces such as the pseudo-force $-ma$ and the Coriolis force must be introduced.
At the highest point of a projectile's trajectory, the vertical component of velocity is zero, but the acceleration is non-zero and equal to $g$ downward.

Glossary

DisplacementA vector quantity representing the shortest straight-line distance from the initial to the final position.

Instantaneous VelocityThe time derivative of the position vector, given by $v = \frac{dr}{dt}$.

Horizontal RangeThe total horizontal distance covered by a projectile during its flight, given by $R = \frac{u^2 \sin(2\theta)}{g}$.

Coriolis AccelerationAn acceleration component experienced by particles in a rotating non-inertial frame, given by $a_c = -2(\omega \times v)$.

Distance vs Displacement

Parameter	Distance	Displacement
Nature	Scalar quantity	Vector quantity
Dependence	Depends on the actual path taken	Depends only on initial and final positions
Magnitude	Always positive or zero	Can be positive, negative, or zero

Inertial vs Non-Inertial Frames

Parameter	Inertial Frame	Non-Inertial Frame
Newton's Laws	Valid without modification	Valid only when pseudo-forces are introduced
Acceleration	Zero acceleration or moving with constant velocity	Accelerating or rotating

Practice Quiz

Choose an option to reveal the explanation.

1. What are the dimensions of displacement?

Explanation: Displacement represents a straight-line distance, hence its dimension is length, denoted as $[L]$.

2. Which of the following represents the instantaneous velocity vector?

Explanation: Instantaneous velocity is defined mathematically as the time derivative of the position vector $r(t)$, expressed as $v = \frac{dr}{dt}$.

3. The area under a velocity-time graph represents which of the following quantities?

Explanation: Integrating the velocity with respect to time gives the total displacement of the particle over the given time interval.

4. At what launch angle is the maximum horizontal range achieved for a given initial speed?

Explanation: The range formula $R = \frac{u^2 \sin(2\theta)}{g}$ reaches its maximum value when $\sin(2\theta) = 1$, which corresponds to $2\theta = 90^\circ$ or $\theta = 45^\circ$.

5. What is the shape of the trajectory of a projectile in a uniform gravitational field without air resistance?

Explanation: Due to independent horizontal motion with constant velocity and vertical motion under constant acceleration, the path is strictly parabolic.

6. What is the relative velocity of body A with respect to body B when their velocities are $v_A$ and $v_B$?

Explanation: The relative velocity of body A with respect to body B is vectorially given by $v_{AB} = v_A - v_B$.

7. Which force must be introduced in a rotating non-inertial frame alongside pseudo-forces?

Explanation: In rotating frames, the Coriolis force $a_c = -2(\omega \times v)$ must be introduced to properly apply equations of motion.

8. What is the correct expression for the Coriolis acceleration in a rotating non-inertial frame?

Explanation: Coriolis acceleration is defined vectorially by $a_c = -2(\omega \times v)$, where $\omega$ is the angular velocity and $v$ is the particle velocity.

9. What is the tangential and normal acceleration for general curvilinear motion in a plane?

Explanation: Curvilinear motion acceleration has a tangential component $\frac{dv}{dt}\hat{t}$ and a normal centripetal component $\frac{v^2}{R}\hat{n}$.

10. Which of the following is true at the highest point of a projectile's trajectory?

Explanation: At the peak, only the vertical component of velocity vanishes; horizontal velocity remains constant, and downward acceleration equals $g$.

11. Under what condition are distance and displacement equal in magnitude?

Explanation: Displacement equals distance only when the particle moves along a straight line without reversing direction.

12. What does free fall near the Earth's surface represent?

Explanation: Free fall represents vertical motion under a constant downward gravitational acceleration $g \approx 9.8 \text{ m/s}^2$.

13. Which transformation relates coordinates between two inertial frames moving with constant relative velocity?

Explanation: Galilean transformation relates position coordinates between two inertial reference frames moving with a constant relative velocity.

14. What is the third equation of motion under constant acceleration?

Explanation: The third kinematic equation is $v^2 = u^2 + 2as$, relating final velocity, initial velocity, acceleration, and displacement.

15. Why do all objects fall with identical acceleration under gravity in a vacuum?

Explanation: Newton's second law shows $a = \frac{mg}{m} = g$, which is independent of the mass of the falling object.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. A projectile is launched with an initial speed $u$ at an angle $\theta$ to the horizontal. What is the expression for its horizontal range $R$?

Explanation: The standard horizontal range formula for ground-to-ground projectile motion is $R = \frac{u^2 \sin(2\theta)}{g}$.

2. What is the magnitude of the Coriolis acceleration for a particle moving with velocity $v$ in a frame rotating with angular velocity $\omega$?

Explanation: Vectorially, the Coriolis acceleration is given by $a_c = -2(\omega \times v)$.

3. If a body travels with constant acceleration $a$, initial velocity $u$, and time $t$, its displacement $s$ is given by:

Explanation: The second equation of motion is $s = ut + \frac{1}{2}at^2$.

4. Which of the following kinematic equations is independent of time $t$?

Explanation: The third equation $v^2 = u^2 + 2as$ relates velocity, acceleration, and displacement without involving time $t$.

5. In curvilinear motion, the acceleration vector is resolved into tangential and normal components. What is the normal component of acceleration?

Explanation: The normal (centripetal) component of acceleration for planar curvilinear motion is $\frac{v^2}{R}\hat{n}$.

6. What is the relative velocity $v_{AB}$ of train A moving east at $20 \text{ m/s}$ relative to train B moving east at $15 \text{ m/s}$?

Explanation: $v_{AB} = v_A - v_B = 20 - 15 = 5 \text{ m/s}$ east.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. What is the time derivative of the position vector $r(t)$ called?

Explanation: Instantaneous velocity is defined as the time derivative of the position vector $r(t)$.

2. Which of the following statements about projectile motion is correct?

Explanation: In the absence of air resistance, horizontal acceleration is zero, vertical acceleration is $g$, and the trajectory is parabolic.

3. What fictitious force appears in a linear non-inertial frame accelerating with $a$?

Explanation: For a linear frame accelerating with $a$, the pseudo-force introduced is $-ma$.

4. If a particle's position is given by $r(t) = t^2 \hat{i} + t \hat{j}$, what is its velocity vector at $t = 2 \text{ s}$?

Explanation: Velocity $v = \frac{dr}{dt} = 2t\hat{i} + \hat{j}$. At $t = 2$, $v = 4\hat{i} + \hat{j}$.

5. What is the unit of the tangential and normal acceleration components?

Explanation: All acceleration components possess units of meters per second squared ($\text{m/s}^2$).

6. Under constant acceleration, which equation links initial velocity, final velocity, acceleration, and time?

Explanation: The first equation of motion is $v = u + at$.

7. What is the value of acceleration due to gravity $g$ typically used in standard free-fall problems near Earth's surface?

Explanation: Standard downward gravitational acceleration near the Earth's surface is approximately $g \approx 9.8 \text{ m/s}^2$.

8. Which quantity is a vector?

Explanation: Displacement is a vector quantity possessing both magnitude and direction.

9. In Galilean transformations, what parameter connects position coordinates between two inertial frames?

Explanation: Galilean transformation relates coordinates between inertial frames moving with a constant relative velocity.

10. What happens to the horizontal component of velocity in an ideal projectile motion?

Explanation: Since there is zero horizontal acceleration in ideal projectile motion, the horizontal component of velocity remains constant.