Motion In A Straight Line: Complete Class 11 Notes For NEET 2027 by PW

Motion in a straight line deals with the movement of an object along a single dimension. The chapter introduces the quantities used to describe this motion and shows how they are related to one another. Concepts such as reference frame, distance, displacement, speed, velocity and acceleration are essential for understanding one-dimensional motion.

For effective revision, you should also be comfortable applying the equations of motion and interpreting physical quantities from graphs. In this PW Physics revision, you will cover motion under gravity, stopping distance and the slope and area relationships of position-time, velocity-time and acceleration-time graphs.

Rest, Motion, And Reference Frame

Whether an object is at rest or in motion depends on the observer and the reference frame being used.

An object is said to be at rest when its position remains unchanged with respect to an observer. It is said to be in motion when its position changes with respect to that observer.

For example, a passenger sitting inside a moving train is at rest relative to another passenger in the same train. However, the same passenger is in motion relative to a person standing at the station.

Therefore, rest and motion are relative terms.

A reference frame is a coordinate system along with a clock used to describe the position and motion of an object. A room, vehicle, train or the ground can serve as a reference frame depending on the situation.

When describing motion, the observer is considered to be at rest in their own reference frame.

Quick Recall: Whether an object is at rest or in motion depends on the reference frame from which it is observed.

Distance And Displacement

Distance is the total length of the actual path travelled by an object. It is a scalar quantity, so it has magnitude but no direction.

Displacement is the change in position of an object from its initial position to its final position. It is a vector quantity and therefore has both magnitude and direction.

The relation between the two is:

Distance ≥ |Displacement|

When an object moves along a straight line without reversing its direction:

Distance = |Displacement|

If the object returns to its starting position, its displacement becomes zero even though it has covered a non-zero distance.

Distance And Displacement In Circular Motion

For motion along a circle of radius rr through an angle θ\theta in radians, the distance travelled along the arc is:

d=rθd=r\theta

The magnitude of displacement is the chord joining the initial and final positions:

s = 2r sin(θ/2)

Thus, distance follows the curved path, whereas displacement connects the two positions directly.

Vector Form Of Displacement

Suppose an object moves:

• 44 m east

• 33 m north

• 1212 m upward

Its displacement vector is:

s = 4i + 3j + 12k

Its magnitude is:

|s| = √(4² + 3² + 12²) = 13 m

The total distance travelled is:

4 + 3 + 12 = 19 m

Speed And Velocity

Speed describes how quickly distance is covered. It is the rate of change of distance with time.

v = Distance / Time

Speed is a scalar quantity with SI unit m/s and dimensions [LT⁻¹].

Average Speed

Average speed is calculated using total distance and total time:

v_avg = Total distance / Total time

For equal distances travelled with speeds v₁ and v₂:

v_avg = 2v₁v₂ / (v₁ + v₂)

For equal time intervals with speeds v₁, v₂, …, vₙ:

v_avg = (v₁ + v₂ + … + vₙ) / n

Instantaneous Speed

Instantaneous speed represents the speed of an object at a particular instant:

v = ds/dt

For uniform speed, equal distances are covered in equal intervals of time. In non-uniform motion, the distances covered in equal time intervals are unequal.

Velocity

Velocity is the rate of change of displacement with time:

v = d(s)/dt

Unlike speed, velocity is a vector quantity.

Average velocity is:

v_avg = Total displacement / Total time

Velocity can be positive, negative or zero depending on the chosen direction and the object's motion.

Uniform velocity requires both the magnitude and direction of velocity to remain constant. Therefore, an object moving with uniform velocity must move along a straight line with constant speed.

Instantaneous velocity is directed along the tangent to the path at that instant.

Quick Recall: Speed is based on distance, while velocity is based on displacement.

Acceleration

Acceleration measures how quickly velocity changes with time.

a = dv/dt

Its SI unit is m/s², and its dimensions are [LT⁻²].

Acceleration can occur when:

1. The speed changes.

2. The direction of motion changes.

3. Both speed and direction change.

Average Acceleration

Average acceleration is the change in velocity divided by the time interval:

a_avg = (v_f − v_i) / Δt

For velocity expressed as a function of time:

a = dv/dt

If displacement is given as a function of time:

v = dx/dt

a = d²x/dt²

If velocity is expressed as a function of position:

a = v(dv/dx)

Zero Acceleration

Zero acceleration means that velocity remains constant:

a = 0 ⇒ v = constant

It does not necessarily mean that the object is at rest.

Distance can be obtained by integrating velocity with respect to time, while velocity can be obtained by integrating acceleration with respect to time.

Equations Of Motion

For motion with constant acceleration, the following equations are used:

First Equation

v = u + at

Second Equation

s = ut + ½at²

Third Equation

v² = u² + 2as

Fourth Equation

s = [(u + v)/2]t

Here:

• u = initial velocity

• v = final velocity

• a = constant acceleration

• t = time

• s = displacement

These equations are applicable only when acceleration is constant. The signs of u, v, a and s should be assigned according to the chosen coordinate direction.

NEET Tip: Before selecting an equation, identify the quantities given in the question and the quantity that needs to be calculated.

Stopping Distance

Stopping distance is the distance covered by an object from the instant braking begins until it comes to rest.

During braking, acceleration acts opposite to the direction of velocity. If the final velocity is zero, the third equation of motion gives:

0 = u² − 2as

Therefore:

s = u²/(2a)

Here, a represents the magnitude of retardation.

The important relationship is:

s ∝ u²

Therefore, if the initial speed is doubled, the braking distance becomes four times the original value.

Reaction Distance

A moving vehicle also travels some distance during the driver's reaction time t.

Reaction distance:

s_r = ut

Hence, total stopping distance is:

s_total = ut + u²/(2a)

Using the relation F = ma, braking distance can also be written as:

s = u²m/(2F)

Motion Under Gravity

When air resistance is neglected, a freely moving object near Earth's surface experiences approximately constant acceleration due to gravity.

Taking upward as positive:

a = −g

Upward Projection

For an object projected vertically upward with initial speed u:

v = u − gt

At the highest point, v = 0.

The maximum height reached is:

h_max = u²/(2g)

Time taken to reach maximum height:

t_ascent = u/g

For a body returning to the same level, the total time of flight is:

T = 2u/g

At the highest point, the velocity becomes zero momentarily, but gravitational acceleration continues to act downward:

a = −g

Object Dropped From Rest

For an object falling freely from rest through a height h:

v = √(2gh)

The time taken is:

t = √(2h/g)

The distances covered in successive equal intervals of time follow:

1 : 3 : 5 : 7 : …

Object Thrown Downward

If an object is thrown downward from height h with initial speed u, its impact speed is:

v_impact = √(u² + 2gh)

Object Thrown Upward From A Building

If an object is projected upward from the top of a building of height h, the maximum height above the ground is:

H_max = h + u²/(2g)

Its speed when it reaches the ground is:

v_impact = √(u² + 2gh)

If an object is released from a moving frame, it retains the velocity of that frame at the instant of release.

Quick Recall: At the highest point of vertical motion, velocity is zero, but acceleration due to gravity is still g downward.

Motion Graphs

Graphs provide a visual way to understand how position, velocity and acceleration change with time.

Position-Time Graph

For an x-t graph, the slope gives velocity.

• Positive slope → positive velocity

• Zero slope → zero velocity

• Negative slope → negative velocity

• Increasing positive slope → positive acceleration

• Positive but decreasing slope → decreasing velocity

The tangent to the position-time curve at a point gives the instantaneous velocity at that point.

Velocity-Time Graph

The slope of a v-t graph gives acceleration:

a = dv/dt

The area under the velocity-time graph gives displacement.

• Area above the time axis → positive displacement

• Area below the time axis → negative displacement

For total distance, the magnitudes of the areas on both sides of the time axis are added.

Acceleration-Time Graph

The area under an a-t graph represents the change in velocity:

Δv = ∫a dt

Therefore:

Area under a-t graph = Δv

Acceleration-Displacement Relation

For velocity expressed as a function of displacement:

a = v(dv/dx)

Therefore:

∫a dx = (v_f² − v_i²)/2

The area under an a-x graph is consequently related to the change in v²/2.

Quick Revision Points

1. Rest and motion depend on the reference frame.

2. Distance is a scalar, while displacement is a vector.

3. Distance is always greater than or equal to the magnitude of displacement.

4. A body returning to its starting point has zero displacement but non-zero distance.

5. Speed is based on distance; velocity is based on displacement.

6. Zero acceleration means constant velocity, not necessarily zero velocity.

7. The instantaneous velocity is tangent to the path.

8. The equations of motion apply when acceleration is constant.

9. Stopping distance varies as the square of initial speed.

10. At the highest point of vertical motion, velocity is zero but acceleration is −g-g when upward is taken as positive.

11. Distances covered in successive equal intervals during free fall from rest follow 1:3:5:7...

12. The slope of an x−tx-t graph gives velocity.

13. The slope of a v−tv-t graph gives acceleration.

14. The area under a v−tv-t graph gives displacement.

15. The area under an a−ta-t graph gives change in velocity.

16. The area under a v−tv-t graph must be treated with signs for displacement.

17. Total distance from a v−tv-t graph is obtained by adding the magnitudes of the relevant areas.

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