Motion - Class 11 Physics

Motion in a Straight Line

Overview: Kinematics deals with the description of motion without considering its causes. This chapter covers rectilinear motion, graphical analysis, and equations of motion.

1. Frame of Reference

A system of coordinates/axes attached to an observer having a clock with him, with respect to which the observer can describe position, displacement, velocity etc., of a moving object.

Inertial Frame: Need Newton's laws to hold true (Non-accelerating).
Non-Inertial Frame: Accelerating frames (Pseudo forces act).

Point Object: If the size of the object is much smaller than the distance it travels.

2. Uniform & Non-Uniform Motion

Uniform Motion

If an object covers equal distances in equal intervals of time, however small these intervals may be.

Velocity is constant.
Acceleration is zero.
x-t graph is a straight line inclined to time axis.

Non-Uniform Motion

If an object covers unequal distances in equal intervals of time.

Velocity changes with time.
Acceleration is non-zero.

3. Elementary Differentiation & Integration

Instantaneous Velocity: Limit of average velocity as time interval approaches zero.

v = lim_(Δt→0) (Δx / Δt) = dx/dt

Instantaneous Acceleration: Rate of change of velocity.

a = dv/dt = d²x/dt²

Integration (Reverse Process): Used to find displacement from velocity or velocity from acceleration.

x = ∫ v dt

v = ∫ a dt

4. Velocity-Time & Position-Time Graphs

Position-Time (x-t) Graph

Slope: Gives Velocity (dx/dt).
Straight line: Uniform motion.
Curved line: Accelerated motion.
Parallel to time axis: Rest.

Velocity-Time (v-t) Graph

Slope: Gives Acceleration (dv/dt).
Area under curve: Gives Displacement.
Straight line: Uniform acceleration.

5. Uniformly Accelerated Motion

If the velocity of an object changes by equal amounts in equal intervals of time.

For uniform acceleration 'a' (Graphical Treatment):

First Equation of Motion

v = u + at

Derived from definition of acceleration (Slope of v-t graph).

Second Equation of Motion

s = ut + ½at²

Derived from Area under v-t graph (Area of rectangle + Area of triangle).

Third Equation of Motion

v² - u² = 2as

Derived from Area of Trapezium under v-t graph.

Stopping Distance: Distance traveled before coming to rest.

d_s = u² / 2a

6. Relative Velocity

Velocity of object A with respect to object B.

v_AB = v_A - v_B

If moving in same direction: subtract speeds.
If moving in opposite direction: add speeds.

Numericals - Motion in a Straight Line

Numericals

Average Velocity

Q1. A car travels from point A to B at 30 km/h and returns from B to A at 20 km/h. Find average speed and average velocity.

Distance A to B = B to A = x km.

Time t₁ = x / 30

Time t₂ = x / 20

Average Speed:

V_avg = Total Distance / Total Time

V_avg = 2x / (x/30 + x/20)

V_avg = 2 / (1/30 + 1/20)

V_avg = 2 / [(2+3)/60]

V_avg = (2 × 60) / 5

V_avg = 120 / 5 = 24 km/h

Average Velocity:

Displacement is zero (returns to start).

Velocity_avg = 0 km/h

Equations of Motion

Q2. A car moving at 10 m/s accelerates constantly at 2 m/s². How much distance does it cover in 10 seconds?

Given: u = 10 m/s, a = 2 m/s², t = 10 s

s = ut + ½at²

s = (10 × 10) + ½ × 2 × (10)²

s = 100 + (1 × 100)

s = 100 + 100

s = 200 m

Calculus - Differentiation

Q3. The displacement of a particle is given by x = 3t² + 2t + 5 meters. Find instantaneous velocity at t = 2s.

Velocity v = dx/dt

v = d/dt (3t² + 2t + 5)

v = 3(2t) + 2(1) + 0

v = 6t + 2

At t = 2s:

v = 6(2) + 2

v = 12 + 2

v = 14 m/s

Calculus - Integration

Q4. Acceleration of a particle is a = 2t m/s². If initial velocity is 5 m/s, find velocity at t=3s.

a = dv/dt → dv = a dt

∫ dv = ∫ 2t dt

v = t² + C

At t=0, v=5 m/s (Initial Condition)

5 = 0² + C → C = 5

Equation: v = t² + 5

At t=3s:

v = (3)² + 5

v = 9 + 5

v = 14 m/s

Free Fall

Q5. A ball is dropped from a tower of height 80m. Take g = 10 m/s². Find the time taken to reach the ground.

u = 0 (dropped), s = 80 m, a = g = 10 m/s²

s = ut + ½at²

80 = 0 + ½ × 10 × t²

80 = 5t²

t² = 16

t = 4 s

nth Second Distance

Q6. Find ratio of distances derived in 3rd second and 3 seconds starting from rest.

Distance in nth second: S_n = u + a/2 (2n - 1)

Distance in n seconds: S = ut + ½at²

Given u = 0.

In 3rd second (n=3):

S_3rd = 0 + a/2 (2×3 - 1) = 5a/2

In 3 seconds (t=3):

S₃ = 0 + ½a(3)² = 9a/2

Ratio:

Ratio = (5a/2) : (9a/2)

Ratio = 5 : 9

Stopping Distance

Q7. A car moving at 20 m/s is stopped by applying brakes which produce retardation of 4 m/s². Find stopping distance.

u = 20 m/s, v = 0, a = -4 m/s²

v² - u² = 2as

0 - (20)² = 2 × (-4) × s

-400 = -8s

s = 400 / 8

s = 50 m

Relative Velocity

Q8. Two trains A and B of lengths 100m and 150m are moving in opposite directions at 54 km/h and 72 km/h. Time to cross?

Opposite direction: Relative Speed adds up.

V_rel = 54 + 72 = 126 km/h

V_rel = 126 × (5/18) = 7 × 5 = 35 m/s

Total Distance to cover = Sum of lengths

D = 100 + 150 = 250 m

Time = Distance / Speed

t = 250 / 35

t = 50 / 7

t ≈ 7.14 s

Graph Analysis

Q9. Velocity-Time graph is a straight line passing through origin with coordinates (4s, 20m/s). Find displacement in 4s.

Displacement = Area under v-t graph.

Graph forms a triangle with base 4s and height 20m/s.

Area = ½ × base × height

Displacement = ½ × 4 × 20

Displacement = 40 m

Reaction Time

Q10. A driver takes 0.2s to apply brakes (reaction time). If he is moving at 54 km/h, what distance does he travel during reaction time?

During reaction time, velocity is constant.

u = 54 km/h = 15 m/s

t = 0.2 s

Distance = Speed × Time

d = 15 × 0.2

d = 3.0 m

Formulas & Facts - Motion

Equations & Formulas

Concept	Formula
Average Speed	Total Distance / Total Time
Instantaneous Velocity	v = dx/dt
Instantaneous Acc.	a = dv/dt = d²x/dt²
1st Eq of Motion	v = u + at
2nd Eq of Motion	s = ut + ½at²
3rd Eq of Motion	v² - u² = 2as
Dist. in nth second	S_n = u + a/2 (2n - 1)
Relative Velocity	v_AB = v_A - v_B
Max Height (Projected Up)	H = u² / 2g
Time of Flight (Up)	T = 2u / g

50 NEET Facts

Key points for Motion in a Straight Line.

1. Path Length vs Displacement Path length is the actual distance traveled and is a scalar. Displacement is the shortest distance between initial and final points and is a vector. Path length ≥ |Displacement|.

2. Average Velocity Zero If an object returns to its starting point, its average velocity is zero, but average speed is non-zero.

3. Instantaneous Speed Instantaneous speed is the magnitude of instantaneous velocity. Unlike average values, at a specific instant, |Velocity| always equals Speed.

4. Negative Acceleration Acceleration is negative if velocity is decreasing in the positive direction (retardation), or velocity is increasing in the negative direction.

5. Area under v-t Graph The area under the velocity-time graph gives the displacement. If area is below the time axis, it represents negative displacement. The sum of absolute areas gives total distance.

6. Slope of x-t Graph The slope of the tangent to the position-time curve at any point gives the instantaneous velocity at that point.

7. Slope of v-t Graph The slope of the tangent to the velocity-time curve gives the instantaneous acceleration.

8. Concavity of x-t Graph If x-t graph is concave up (bowl shape), acceleration is positive. If concave down (inverted bowl), acceleration is negative.

9. Stopping Distance Stopping distance is proportional to the square of initial velocity (d ∝ u²). Doubling the speed makes stopping distance 4 times.

10. Motion under Gravity For free fall, a = g (approx 9.8 m/s²) downwards. Equations of motion apply with a = -g (if up is positive).

11. Velocity at Max Height When an object is thrown vertically upwards, its velocity at the highest point is zero, but acceleration is still 'g' downwards.

12. Time Symmetry Time taken to go up equals time taken to fall down to the same level (neglecting air resistance).

13. Speed Symmetry Speed at any point during ascent equals speed at the same point during descent (neglecting air resistance).

14. Relative Velocity Paradox Rain falling vertically appears to fall at an angle to a moving observer. tan(θ) = v_obs / v_rain.

15. Crossing a River To cross a river in minimum time, swimmmer should head perpendicular to river flow. Time = Width / Velocity of swimmer.

16. Shortest Path River To cross a river via shortest path (straight across), swimmer must head upstream at an angle sin(θ) = v_river / v_swimmer.

17. Reaction Time The time interval between seeing an event and applying the brakes/actuation. Distance moved = v × reaction time.

18. Odd Number Rule Distances traveled by a freely falling body in equal successive time intervals are in the ratio 1:3:5:7... (Galileo's Law).

19. Body from moving platform A body dropped from a moving train has the initial velocity of the train but no initial acceleration (except gravity). path is parabolic for ground observer.

20. Acceleration of Rest An object can have zero velocity but non-zero acceleration (e.g., at max height of vertical throw, or simple harmonic motion turning point).

21. Constant Speed, Variable Velocity Possible in circular motion. Velocity direction changes, magnitude stays same. Acceleration is non-zero.

22. Variable Speed, Constant Velocity Impossible. If speed changes, magnitude of velocity changes, so velocity cannot be constant.

23. Zero Acceleration graph Position-time graph is a straight line. Velocity-time graph is a horizontal line.

24. Juggler Problem For n balls in air, time of flight for each ball = n × time interval between throws.

25. Meeting Time Relative distance / Relative velocity. Useful for two trains crossing or police chasing thief.

26. Averages with different Distances If distances are equal, v_avg = 2v1v2 / (v1+v2) (Harmonic Mean).

27. Averages with different Times If time intervals are equal, v_avg = (v1+v2) / 2 (Arithmetic Mean).

28. Distance in last second If a body drops from height h, distance in last second depends on h.

29. Graph Intersection Intersection of two position-time graphs indicates the time and position where two objects meet.

30. Velocity depends on root x If v ∝ √x, then motion is uniformly accelerated. (Since v² = kx, diff w.r.t t gives a = constant).

31. Bullet penetration If a bullet loses 1/n of its velocity passing through a plank, it will enter (n² / (2n-1)) planks before stopping.

32. Tap water drops If drops fall at regular intervals, their positions at any instant follow the square law (1:4:9...).

33. x vs t graph impossible features Graph cannot become vertical (infinite velocity). Graph cannot loop back (two positions at one time). Speed cannot be negative.

34. Relative Acc. Free Fall Acceleration of one falling body relative to another is zero (both have g). Relative velocity increases linearly with time.

35. Balloon Problem A stone dropped from a rising balloon has initial upward velocity equal to balloon's velocity.

36. Parachute First accelerates downwards (g), then opens chute (large retardation), then reaches terminal velocity (a=0).

37. Integration Area Area under a-t graph gives Change in Velocity (not just velocity).

38. Staircase approximation Using small steps of constant velocity to approximate variable velocity is essentially integration (Riemann sum).

39. Overtaking Condition for overtaking: Position of chaser = Position of target at same time t.

40. Closest Distance For limits/closest approach problems, use relative velocity concept. Velocity of approach becomes zero at closest distance.

41. 1D vs 2D Straight line motion is 1D. If acceleration is not parallel/anti-parallel to velocity, it becomes 2D (curved path).

42. Vector nature Equations of motion v=u+at etc are vector equations. Signs (+/-) are crucial for 1D.

43. Displacement ratio In time T, 2T, 3T... starting from rest, displacements are 1:4:9. (Square of time).

44. Velocity ratio In time T, 2T, 3T... starting from rest, velocities are 1:2:3.

45. Same height t1, t2 If a projectile passes same height h at t1 and t2, then T (flight time) = t1+t2 and h = ½g t1 t2.

46. Average Acceleration Change in velocity vector / Total time.

47. Constant Jerk Rate of change of acceleration is Jerk. Not usually covered in basic kinematics but implies cubic position-time relation.

48. Bridge Problem Sound of splash heard after stone drops = Time to fall + Time for sound to travel up.

49. Inclined Plane Acceleration down a smooth incline is g sin(θ).

50. Review of Slope Angle < 90 deg → Positive slope. Angle> 90 deg → Negative slope.

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