Motion in a Straight Line notes

Quick revision8

Formulas5

Average Velocity

v_{avg} = Δx / Δt

Average rate of change of position during a time interval.

v_avg: Average velocity
Δx: Displacement or change in position
Δt: Time interval

Average Speed

\text{Average speed }=\text{ total distance }/\text{ total time}

Total path length covered per unit time.

total distance: Complete length of actual path travelled
total time: Time taken for the journey

Average Acceleration

a_{avg} = Δv / Δt

Average rate of change of velocity.

a_avg: Average acceleration
Δv: Change in velocity
Δt: Time interval

Diagrams3

Position, Distance & Displacement

8 min

To study straight-line motion, we choose a reference point called origin and assign positions along a coordinate axis. In one dimension, position is represented by a coordinate such as x = +5 m or x = -3 m, where the sign shows direction from the origin. Distance is the total path length actually travelled, so it is a scalar and never negative. Displacement is the change in position from initial to final point, so it is a vector and can be positive, negative or zero. Distance depends on the path followed, but displacement depends only on initial and final positions. This distinction is a common NEET conceptual trap.

Key points5

Sign convention must be fixed before solving; usually right/up is positive and left/down is negative.
A particle can have zero displacement even after covering a non-zero distance.
For motion without reversal in one direction, distance equals magnitude of displacement.
Average displacement over several trips is net change in position, not total path length divided by number of trips.
Position vector in 1D is x i-hat, but most NEET questions use signed x-coordinate directly.

Memory tricks2

Distance vs Displacement: Distance = road travelled. Displacement = result of journey.

Sign Rule: Choose a positive direction first; then every position, displacement, velocity and acceleration follows that sign convention.

Examples3

Straight-Line Example

A student moves from x = 2 m to x = 9 m. Displacement = 9 - 2 = +7 m and distance = 7 m.

Return Example

A runner goes 100 m forward and 100 m back. Distance = 200 m, displacement = 0.

Practice MCQ

A particle moves from -3 m to +5 m and then to +1 m. Distance = 8 + 4 = 12 m; displacement = 1 - (-3) = +4 m.

Common mistakes3

Using Distance Instead of Displacement

For average velocity, use displacement, not total distance.

Forgetting Negative Position

A negative coordinate does not mean negative distance; it means position on the negative side of origin.

Assuming Displacement Always Equals Distance

They are equal only when the object moves in a straight line without changing direction.

Reference tables2

Quick revision8

Formulas4

Displacement

Δx = x2 - x1

Net change in position from initial coordinate to final coordinate.

Δx: Displacement
x2: Final position
x1: Initial position

Average Displacement for Multiple Segments

\text{Net displacement }= x_{\text{final}} - x_{\text{initial}}

Even if a body follows many segments, displacement depends only on final and initial positions.

x_final: Final coordinate
x_initial: Initial coordinate

Diagrams3

Speed & Velocity

8 min

Speed and velocity describe how fast an object moves, but they are not identical. Speed is the rate of covering distance and is a scalar, so it has magnitude only. Velocity is the rate of change of displacement and is a vector, so direction matters. Average speed depends on total distance and total time, while average velocity depends on displacement and total time. Instantaneous speed or velocity refers to the value at a particular instant, like a speedometer reading. Uniform motion means equal displacements in equal time intervals, giving constant velocity. Variable motion means velocity changes with time, either in magnitude, direction or both.

Key points5

Speed can never be negative, but velocity can be positive, negative or zero.
If a body returns to its starting point, average velocity is zero but average speed is non-zero.
In one-dimensional motion without direction reversal, average speed equals magnitude of average velocity.
Instantaneous velocity is the slope of position-time graph at that instant.
Uniform speed does not always imply uniform velocity in general motion, but in straight-line motion with fixed direction it does.

Memory tricks2

Speed vs Velocity: Speed uses Distance; Velocity uses Displacement. Remember S-D and V-D.

Average Speed Shortcut: Equal distance means harmonic mean; equal time means arithmetic mean.

Examples3

Numerical Example

A body travels 100 m in 10 s and returns 100 m in 20 s. Average speed = 200/30 = 6.67 m/s; average velocity = 0/30 = 0.

Equal Distance Shortcut

For equal distances at 20 m/s and 30 m/s, average speed = 2 × 20 × 30 / 50 = 24 m/s.

Practice MCQ

Can average speed be zero for a moving body? No. If distance is covered, average speed is positive.

Common mistakes3

Averaging Speeds Directly Every Time

Average speed is not always simple average of speeds. Use total distance divided by total time unless equal time is given.

Ignoring Direction in Velocity

A car moving left with speed 20 m/s may have velocity -20 m/s if right is positive.

Using Distance for Average Velocity

Average velocity must use displacement. For a round trip, it is zero.

Reference tables2

Definition	Distance/time	Displacement/time
Type	Scalar	Vector
Sign	Never negative	Can be positive, negative or zero
Direction	Not included	Included by sign in 1D

Uniform motion	Equal displacements in equal time intervals	Straight line x-t graph
Variable motion	Unequal displacements in equal time intervals	Curved x-t graph
Uniform speed	Equal distances in equal intervals	Constant speed-time graph
Variable speed	Speed changes with time	Changing speed-time graph

Quick revision8

Formulas5

Average Speed

v_{\text{speed}}_{avg} =\text{ total distance }/\text{ total time}

Measures how fast total path length is covered.

v_speed_avg: Average speed
total distance: Total path length travelled
total time: Total time taken

Average Velocity

v_{avg} = (x2 - x1) / (t2 - t1)

Measures rate of change of position over a finite time interval.

v_avg: Average velocity
x2 - x1: Displacement
t2 - t1: Time interval

Instantaneous Velocity

v = dx / dt

Velocity at a particular instant; slope of tangent to position-time graph.

v: Instantaneous velocity
dx: Infinitesimal displacement
dt: Infinitesimal time interval

Diagrams3

Acceleration

8 min

Acceleration measures how quickly velocity changes with time. Since velocity is a vector, acceleration can occur due to change in magnitude or direction, but in straight-line motion direction is represented by sign. Average acceleration is change in velocity divided by time interval, while instantaneous acceleration is the value at a particular instant. Positive acceleration means acceleration is along the chosen positive direction, not necessarily speeding up. Negative acceleration, often called retardation when it opposes motion, may slow a body down. Uniform acceleration is constant acceleration, where velocity changes equally in equal time intervals. Non-uniform acceleration changes with time and is represented by a curved velocity-time graph.

Key points6

Acceleration can be negative even when the object is speeding up in the negative direction.
A body moving with constant velocity has zero acceleration.
Uniform acceleration gives a straight-line velocity-time graph.
Non-uniform acceleration gives a curved velocity-time graph.
Deceleration is not always the same as negative acceleration; it means speed is decreasing.
Free fall near Earth is uniformly accelerated motion with acceleration g downward.

Memory tricks2

Acceleration Sign: Same sign of v and a means speeding up; opposite signs mean slowing down.

Graph Slope: For v-t graph, uphill means positive acceleration, flat means zero acceleration, downhill means negative acceleration.

Examples3

Average Acceleration Example

A car changes velocity from 10 m/s to 30 m/s in 5 s. Acceleration = (30 - 10)/5 = 4 m s^-2.

Negative Acceleration Example

If right is positive and a car moving right slows from 20 m/s to 5 m/s in 3 s, acceleration = -5 m s^-2.

Practice MCQ

A body has v = -10 m/s and a = -2 m/s^2. Its speed increases because velocity and acceleration have the same sign.

Common mistakes3

Calling Every Negative Acceleration Retardation

Negative acceleration is retardation only when it opposes velocity and decreases speed.

Confusing Zero Velocity with Zero Acceleration

At the highest point of vertical motion, velocity is zero but acceleration is still g downward.

Ignoring Units

Acceleration has unit m s^-2, not m s^-1.

Reference tables3

Positive acceleration	Acceleration along positive axis	Car speeding up to the right
Negative acceleration	Acceleration along negative axis	Car accelerating to the left
Retardation	Acceleration opposite to velocity	Brakes applied to a moving car
Uniform acceleration	Constant acceleration	Freely falling body near Earth

Positive	Positive	Increases
Positive	Negative	Decreases
Negative	Positive	Decreases
Negative	Negative	Increases

If velocity is constant, acceleration is?	Zero
SI unit of acceleration?	m s^-2
Slope of v-t graph gives?	Acceleration
Can acceleration be negative while speed increases?	Yes, if velocity is negative

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Formulas3

Average Acceleration

a_{avg} = (v2 - v1) / (t2 - t1)

Change in velocity per unit time over an interval.

a_avg: Average acceleration
v2: Final velocity
v1: Initial velocity
t2 - t1: Time interval

Instantaneous Acceleration

a = dv / dt

Acceleration at an exact instant; slope of tangent on velocity-time graph.

a: Instantaneous acceleration
dv: Small change in velocity
dt: Small time interval

Diagrams3

Kinematic Equations

11 min

Kinematic equations connect displacement, initial velocity, final velocity, acceleration and time for motion with constant acceleration. They are among the most frequently used formulas in NEET Physics. The first equation comes from the definition of acceleration, the second from displacement as average velocity multiplied by time or integration, and the third eliminates time. These equations apply only when acceleration is constant. Free fall is a special case of uniform acceleration where acceleration is g downward. In one-dimensional relative motion, positions and velocities are compared by subtraction, which helps solve chasing, meeting and separation problems. Correct sign convention is the key to avoiding mistakes.

Key points6

Choose positive direction before substituting values.
If upward is positive in free fall, acceleration is -g.
If downward is positive in free fall, acceleration is +g.
At maximum height in vertical motion, velocity becomes zero but acceleration remains g downward.
Equation without time, v^2 = u^2 + 2as, is best when time is not given.
Relative speed for objects moving toward each other is sum of speeds; same direction is difference of speeds.

Memory tricks3

SUVAT: Remember variables as SUVAT: s displacement, u initial velocity, v final velocity, a acceleration, t time.

No Time Formula: When time is absent, think of the square equation: v^2 = u^2 + 2as.

Free Fall Sign: Gravity always points downward; only its sign changes with your chosen positive direction.

Examples4

Solved NEET Numerical

A car starts from rest and accelerates at 2 m s^-2 for 5 s. v = u + at = 0 + 2×5 = 10 m/s. s = ut + 1/2 at^2 = 0 + 1/2×2×25 = 25 m.

Equation Without Time

A body moving at 10 m/s accelerates at 3 m s^-2 through 50 m. v^2 = 100 + 2×3×50 = 400, so v = 20 m/s.

Relative Motion Basic

Two cars move in the same direction at 30 m/s and 20 m/s. Relative speed = 10 m/s. If 100 m apart, catch-up time = 100/10 = 10 s.

Free Fall Example

A stone dropped from rest for 2 s falls h = 1/2 gt^2 = 1/2 × 9.8 × 4 = 19.6 m.

Common mistakes4

Applying Equations to Non-Uniform Acceleration

The standard kinematic equations are valid only for constant acceleration.

Wrong Sign of Acceleration

If upward is positive, g must be written as -9.8 m s^-2.

Confusing Distance with Displacement

The s in equations of motion is displacement, not necessarily total distance.

Assuming Acceleration Zero at Highest Point

At maximum height, velocity is zero but acceleration remains downward g.

Reference tables3

u, a, t given; find v	v = u + at	Direct velocity-time relation
u, a, t given; find s	s = ut + (1/2)at^2	Direct displacement-time relation
u, a, s given; find v	v^2 = u^2 + 2as	Time not needed
u, v, t given; find s	s = [(u + v)/2]t	Uses average velocity

Upward positive	a = -g	Initial velocity positive
Downward positive	a = +g	Initial velocity negative if thrown upward
At highest point	a = downward	v = 0 only at instant
Dropped from rest	u = 0	Acceleration acts downward

Using formulas for variable acceleration	Use only when acceleration is constant
Taking g always positive	Assign sign according to chosen axis
Using distance instead of displacement	Kinematic s is displacement
Forgetting v = 0 at maximum height	Only velocity is zero; acceleration is not zero

Quick revision8

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Formulas6

First Equation of Motion

v = u + at

Derived from a = (v - u)/t for constant acceleration.

v: Final velocity
u: Initial velocity
a: Constant acceleration
t: Time

Second Equation of Motion

s = ut + (1/2)at^2

Gives displacement when initial velocity, acceleration and time are known.

s: Displacement
u: Initial velocity
a: Constant acceleration
t: Time

Third Equation of Motion

v^2 = u^2 + 2as

Equation without time, useful when time is not mentioned.

v: Final velocity
u: Initial velocity
a: Constant acceleration
s: Displacement

Diagrams3

Motion Graphs

10 min

Motion graphs convert equations and observations into visual form. A position-time or displacement-time graph shows how location changes with time; its slope gives velocity. A distance-time graph has non-negative slope because distance never decreases. A velocity-time graph shows how velocity changes; its slope gives acceleration and the area under the graph gives displacement. A speed-time graph gives distance from area because speed is always non-negative. An acceleration-time graph gives change in velocity from the area under the curve. NEET graph questions are usually solved by two tools: slope and area. Correct interpretation of sign, intercept, curvature and units is essential.

Key points7

A horizontal position-time graph means the object is at rest.
A straight inclined position-time graph means uniform velocity.
A curved position-time graph means non-uniform velocity.
A horizontal velocity-time graph means zero acceleration and constant velocity.
Velocity below time-axis represents negative displacement contribution.
Speed-time graph never goes below time-axis.
For graph-based numericals, always check axis units before calculating slope or area.

Memory tricks3

Slope-Area Rule: Remember: Slope shows what it becomes next; area shows what it builds up.

x-v-a Graph Chain: x-t slope gives v; v-t slope gives a; a-t area gives change in v.

Distance vs Displacement Area: v-t area is signed displacement; speed-time area is total distance.

Examples4

Graph-Based Numerical

If velocity increases uniformly from 0 to 20 m/s in 5 s, displacement = area of triangle = 1/2 × 5 × 20 = 50 m.

Slope Example

If position changes from 2 m to 14 m in 3 s on a straight x-t graph, velocity = slope = 12/3 = 4 m/s.

Acceleration-Time Example

If acceleration is 3 m s^-2 for 4 s, change in velocity = area = 3 × 4 = 12 m/s.

Previous NEET-Style Question

A v-t graph lies above the axis for 4 s and below for 2 s. Net displacement is positive area minus negative-side magnitude, not total area.

Common mistakes4

Taking Area Under x-t Graph

Area under position-time graph is not displacement in standard NEET kinematics.

Ignoring Negative Area

Area below the time-axis in a velocity-time graph is negative displacement.

Thinking Distance-Time Graph Can Decrease

Distance travelled cannot decrease with time, so distance-time graph cannot have negative slope.

Confusing Speed-Time and Velocity-Time Graph

Speed is never negative, but velocity can be below the time-axis.

Reference tables3

Position-time	Velocity	No standard kinematic use
Displacement-time	Velocity	No standard kinematic use
Distance-time	Speed	No standard kinematic use
Velocity-time	Acceleration	Displacement

Horizontal line	Rest	Constant velocity
Straight inclined line	Uniform velocity	Uniform acceleration
Curve with increasing slope	Speed increasing	Acceleration changing
Curve with decreasing slope	Speed decreasing	Acceleration changing

Find displacement from v-t graph	Calculate signed area
Find acceleration from v-t graph	Calculate slope
Identify rest from x-t graph	Look for horizontal segment
Compare speeds from x-t graph	Steeper slope means greater speed

Quick revision8

Formulas5

Slope of Position-Time Graph

\text{velocity }= Δx / Δt

Slope of x-t graph gives average velocity; tangent slope gives instantaneous velocity.

Δx: Change in position
Δt: Time interval

Slope of Velocity-Time Graph

\text{acceleration }= Δv / Δt

Slope of v-t graph gives acceleration.

Δv: Change in velocity
Δt: Time interval

Area Under Velocity-Time Graph

\text{displacement }=\text{ area under }v-t\text{ graph}

Signed area above time-axis is positive; below time-axis is negative.

displacement: Net change in position
area: Geometrical area with sign

Diagrams5