Motion in a Straight Line Notes
Study Notes
Topics
5📖 1. Chapter Overview
Overview
Motion in a straight line is the foundation of mechanics and explains how an object changes its position with time along a single axis. To describe motion properly, we first choose a frame of reference, a reference point and a coordinate system. Quantities such as distance and speed are scalars, while displacement, velocity and acceleration are vectors in one dimension with positive or negative signs. This chapter teaches uniform and non-uniform motion, position-time graphs, velocity-time graphs and kinematic equations. For NEET, it is very important because questions are commonly formula-based, graph-based or concept-based and also support later chapters like Laws of Motion, Work-Energy and Gravitation.
- 1Rest and motion are relative; the same object may be at rest for one observer and moving for another.
- 2Sign convention is essential in straight-line motion because direction is represented by positive or negative sign.
- 3A scalar has magnitude only; a vector has magnitude and direction.
- 4Kinematics describes motion without discussing its cause.
- 5NEET often asks direct use of equations, graph interpretation and free fall cases.
- 6Graph questions become easy if slope and area rules are remembered.
Kinematics Order
Remember PVA: Position changes to give Velocity, Velocity changes to give Acceleration.
Graph Shortcut
On motion graphs: slope means rate, area means accumulation.
Real-Life Frame of Reference
A bag kept on a moving train is at rest for a passenger but moving for a person standing on the platform.
Quick MCQ
If a body returns to its starting point after travelling 20 m, its displacement is 0 but distance is 20 m.
Ignoring Reference Frame
Motion cannot be described without saying with respect to what. A passenger is at rest relative to a bus but moving relative to the road.
Confusing Distance and Displacement
Distance is never negative and depends on path; displacement can be positive, negative or zero and depends only on initial and final positions.
Using Speed and Velocity as Always Same
They are equal only in magnitude when motion is in one direction without reversal.
Average rate of change of position during a time interval.
Variables
v_avg=Average velocity
Δx=Displacement or change in position
Δt=Time interval
Total path length covered per unit time.
Variables
total distance=Complete length of actual path travelled
total time=Time taken for the journey
Average rate of change of velocity.
Variables
a_avg=Average acceleration
Δv=Change in velocity
Δt=Time interval
📍 2. Position, Distance & Displacement
Overview
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.
- 1Sign convention must be fixed before solving; usually right/up is positive and left/down is negative.
- 2A particle can have zero displacement even after covering a non-zero distance.
- 3For motion without reversal in one direction, distance equals magnitude of displacement.
- 4Average displacement over several trips is net change in position, not total path length divided by number of trips.
- 5Position vector in 1D is x i-hat, but most NEET questions use signed x-coordinate directly.
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.
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.
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.
Net change in position from initial coordinate to final coordinate.
Variables
Δx=Displacement
x2=Final position
x1=Initial position
Even if a body follows many segments, displacement depends only on final and initial positions.
Variables
x_final=Final coordinate
x_initial=Initial coordinate
⚡ 3. Speed & Velocity
Overview
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.
- 1Speed can never be negative, but velocity can be positive, negative or zero.
- 2If a body returns to its starting point, average velocity is zero but average speed is non-zero.
- 3In one-dimensional motion without direction reversal, average speed equals magnitude of average velocity.
- 4Instantaneous velocity is the slope of position-time graph at that instant.
- 5Uniform speed does not always imply uniform velocity in general motion, but in straight-line motion with fixed direction it does.
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.
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.
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.
Measures how fast total path length is covered.
Variables
v_speed_avg=Average speed
total distance=Total path length travelled
total time=Total time taken
Measures rate of change of position over a finite time interval.
Variables
v_avg=Average velocity
x2 - x1=Displacement
t2 - t1=Time interval
Velocity at a particular instant; slope of tangent to position-time graph.
Variables
v=Instantaneous velocity
dx=Infinitesimal displacement
dt=Infinitesimal time interval
🚀 4. Acceleration
Overview
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.
- 1Acceleration can be negative even when the object is speeding up in the negative direction.
- 2A body moving with constant velocity has zero acceleration.
- 3Uniform acceleration gives a straight-line velocity-time graph.
- 4Non-uniform acceleration gives a curved velocity-time graph.
- 5Deceleration is not always the same as negative acceleration; it means speed is decreasing.
- 6Free fall near Earth is uniformly accelerated motion with acceleration g downward.
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.
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.
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.
Change in velocity per unit time over an interval.
Variables
a_avg=Average acceleration
v2=Final velocity
v1=Initial velocity
t2 - t1=Time interval
Acceleration at an exact instant; slope of tangent on velocity-time graph.
Variables
a=Instantaneous acceleration
dv=Small change in velocity
dt=Small time interval
📐 5. Kinematic Equations
Overview
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.
- 1Choose positive direction before substituting values.
- 2If upward is positive in free fall, acceleration is -g.
- 3If downward is positive in free fall, acceleration is +g.
- 4At maximum height in vertical motion, velocity becomes zero but acceleration remains g downward.
- 5Equation without time, v^2 = u^2 + 2as, is best when time is not given.
- 6Relative speed for objects moving toward each other is sum of speeds; same direction is difference of speeds.
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.
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.
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.
Derived from a = (v - u)/t for constant acceleration.
Variables
v=Final velocity
u=Initial velocity
a=Constant acceleration
t=Time
Gives displacement when initial velocity, acceleration and time are known.
Variables
s=Displacement
u=Initial velocity
a=Constant acceleration
t=Time
Equation without time, useful when time is not mentioned.
Variables
v=Final velocity
u=Initial velocity
a=Constant acceleration
s=Displacement
📊 6. Motion Graphs
Overview
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.
- 1A horizontal position-time graph means the object is at rest.
- 2A straight inclined position-time graph means uniform velocity.
- 3A curved position-time graph means non-uniform velocity.
- 4A horizontal velocity-time graph means zero acceleration and constant velocity.
- 5Velocity below time-axis represents negative displacement contribution.
- 6Speed-time graph never goes below time-axis.
- 7For graph-based numericals, always check axis units before calculating slope or area.
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.
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.
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.
Slope of x-t graph gives average velocity; tangent slope gives instantaneous velocity.
Variables
Δx=Change in position
Δt=Time interval
Slope of v-t graph gives acceleration.
Variables
Δv=Change in velocity
Δt=Time interval
Signed area above time-axis is positive; below time-axis is negative.
Variables
displacement=Net change in position
area=Geometrical area with sign
Formula Sheet
10Average rate of change of position during a time interval.
Variables
v_avg=Average velocity
Δx=Displacement or change in position
Δt=Time interval
Total path length covered per unit time.
Variables
total distance=Complete length of actual path travelled
total time=Time taken for the journey
Average rate of change of velocity.
Variables
a_avg=Average acceleration
Δv=Change in velocity
Δt=Time interval
Relates final velocity, initial velocity, acceleration and time for constant acceleration.
Variables
v=Final velocity
u=Initial velocity
a=Constant acceleration
t=Time
Gives displacement in time t when acceleration is constant.
Variables
s=Displacement
u=Initial velocity
a=Constant acceleration
t=Time
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NEET PYQs — Motion in a Straight Line
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The following plots show variation of velocity (v) with time (t) of a ball thrown vertically upward, and falling back. Which of the following plots is/are correct?
A toy car with charge q moves on a frictionless horizontal plane surface under the influence of a uniform electric field $\vec{E}$. Due to the force q$\vec{E}$, its velocity increases from 0 to 6 m/s in one second. At that instant the direction of the field is reversed. The car continues to move for two more seconds under the influence of this field. The average velocity and the average speed of the toy car between 0 to 3 seconds are:
An object is thrown vertically upward with some speed. It crosses two points p, q which are separated by h metre. If tₚ is the time between p and highest point and coming back, and t$_q$ is the time between q and highest point and coming back, relate acceleration due to gravity g, tₚ, t$_q$ and h.
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