3Motion in a Straight Line

📚
MOTION IN A STRAIGHT LINE
▢ Definition:
❖ Rectilinear Motion: Straight-line motion

Table 1: Condition for Straight Line Motion

Condition
Path
Velocity constant
Straight line
Acceleration and velocity at 0°
Straight line
Acceleration and velocity at 180°
Straight line
▢ Distance and Displacement:

Table 1: Distance vs Displacement

Feature
Distance
Displacement
Definition
Length of actual path followed by moving body
Shortest distance between initial and final position with direction
Quantity
Scalar
Vector
Value
Never negative
Positive, negative or zero
For moving body
Always increases with time
May increase or decrease with time
Relation
\(Distance \geq |Displacement|\)
\(|Displacement| \leq Distance\)
▢ Speed and Velocity:

Table 1: Speed vs Velocity

Feature
Speed
Velocity
Definition
Time rate of change of distance
Time rate of change of displacement
Quantity
Scalar
Vector
Formula
\(Speed = \frac{Distance}{Time}\)
\(Velocity = \frac{Displacement}{Time}\)
Value
Never negative
Positive, negative or zero
Types
Uniform, non-uniform, average, instantaneous
Average, instantaneous
❖ Uniform and Non-uniform Speed:

Table 1: Types of Speed

Type
Meaning
Uniform speed
Equal distances in equal time intervals
Non-uniform / variable speed
Equal distances in unequal times OR unequal distances in equal times
❖ Average and Instantaneous Speed:

Table 1: Speed Formulae

Type
Formula
Defined for
Average speed
\(v_{av}=\frac{Total\ distance}{Total\ time}=\frac{x-x_0}{t-t_0}=\frac{\Delta x}{\Delta t}\)
Period of time
Instantaneous speed
\(v_i=\lim_{\Delta t\to0}\frac{\Delta x}{\Delta t}=\frac{dx}{dt}\)
Point of time
❖ Average and Instantaneous Velocity:

Table 1: Velocity Formulae

Type
Formula
Defined for
Average velocity
\(\vec v_{av}=\frac{Total\ displacement}{Total\ time}=\frac{\vec s-\vec s_0}{t-t_0}=\frac{\Delta \vec s}{\Delta t}\)
Period of time
Instantaneous velocity
\(\vec v_i=\lim_{\Delta t\to0}\frac{\Delta \vec s}{\Delta t}=\frac{d\vec s}{dt}\)
Point of time
❖ Average Velocity for Different Segments:
◉ General: \(v_{av}=\frac{s_1+s_2+s_3+...+s_n}{t_1+t_2+t_3+...+t_n}\)
◉ Using Velocities: \(v_{av}=\frac{s_1+s_2+s_3+...+s_n}{\frac{s_1}{v_1}+\frac{s_2}{v_2}+\frac{s_3}{v_3}+...+\frac{s_n}{v_n}}\)
◉ Equal Displacements:
◈ Condition: \(s_1=s_2=s_3=...=s_n\)
◈ Formula: \(\frac{n}{v_{av}}=\frac{1}{v_1}+\frac{1}{v_2}+\frac{1}{v_3}+...+\frac{1}{v_n}\)
◈ Meaning: Average velocity = Harmonic mean
◈ For Two Velocities: \(v_{av}=\frac{2v_1v_2}{v_1+v_2}\)
◉ Equal Times:
◈ Condition: \(t_1=t_2=t_3=...=t_n\)
◈ Formula: \(v_{av}=\frac{v_1+v_2+v_3+...+v_n}{n}\)
◈ Meaning: Average velocity = Arithmetic mean
◈ For Two Velocities: \(v_{av}=\frac{v_1+v_2}{2}\)
◉ Notes:
  • •
    If \(s\) are distances → formula gives average speed
  • •
    If net displacement = 0 → average velocity = 0
▢ Acceleration:
❖ Definition: Time rate of change of velocity
❖ Quantity: Vector

Table 1: Acceleration Types

Type
Formula / Meaning
Average acceleration
\(\vec a_{av}=\frac{\vec v-\vec v_0}{t-t_0}=\frac{\Delta \vec v}{\Delta t}\)
Instantaneous acceleration
\(\vec a_i=\lim_{\Delta t\to0}\frac{\Delta \vec v}{\Delta t}=\frac{d\vec v}{dt}\)
Uniform acceleration
Velocity changes equally in equal time intervals
Variable acceleration
Velocity changes unequally in equal time intervals
▢ Rest and Motion:

Table 1: Motion Based on Displacement-Time Relation

Condition
Relation
Formula / Point
Rest
\(s\propto t^0\)
Position constant
Uniform motion
\(s\propto t\)
\(s=ut\)
Constant acceleration
\(s\propto t^2\)
Equations of motion apply
Variable acceleration
\(s\propto t^3\)
Non-uniform acceleration
▢ Equations of Motion:
❖ Notations:

Table 1: Symbols

Symbol
Meaning
\(s\)
Displacement
\(s_{nth}\)
Displacement in particular second
\(u\)
Initial velocity
\(v\)
Final velocity
\(t\)
Time
\(a\)
Acceleration
❖ Fundamental Formulae:
  • •
    \(v=u+at\)
  • •
    \(v^2=u^2+2as\)
  • •
    \(s=ut+\frac{1}{2}at^2\)
  • •
    \(s_{nth}=u+\frac{a}{2}(2n-1)\)
❖ Generalizations from Rest:

Table 1: Uniform Acceleration from Rest

Quantity
Ratio
Distance up to ends of successive equal time intervals
\(1:4:9:...:n^2\)
Distance in successive equal time intervals
\(1:3:5:...:(2n-1)\)
Velocity after successive equal time intervals
\(1:2:3:...:n\)
Time to reach successive equally spaced points
\(\sqrt1:\sqrt2:\sqrt3:...:\sqrt n\)
Time to cover successive equal distances
\((\sqrt1-\sqrt0):(\sqrt2-\sqrt1):(\sqrt3-\sqrt2):...:(\sqrt n-\sqrt{n-1})\)
▢ Motion Under Gravity:
❖ Free Fall from Height H:
◉ Conditions:
  • •
    \(u=0\)
  • •
    \(a=g\)
  • •
    \(s=H\)

Table 1: Freely Falling Body

Quantity
Formula
Time to reach ground
\(t=\sqrt{\frac{2H}{g}}\)
Velocity at height \(h\) above ground
\(v_1=\sqrt{2g(H-h)}\)
Striking velocity on ground
\(v=gt=\sqrt{2gH}\)
Average velocity for entire motion
\(\bar v=\frac{gt}{2}=\sqrt{\frac{gH}{2}}\)
Distance covered in \(n^{th}\) second
\(s_{nth}=\frac{g}{2}(2n-1)\)
◉ Distance Ratios:
  • •
    Distance covered in 1, 2, 3 seconds → \(1:4:9:...:n^2\)
  • •
    Distance covered in 1st, 2nd, 3rd seconds → \(1:3:5:...:(2n-1)\)
❖ Vertical Upward Projection:
◉ Conditions:
  • •
    At maximum height: \(v=0\)
  • •
    \(a=-g\)
  • •
    \(s=H\)

Table 1: Upward Projection Formulae

Quantity
Formula
Maximum height
\(H=\frac{u^2}{2g}\)
Time to reach maximum height
\(t=\frac{u}{g}=\sqrt{\frac{2H}{g}}\)
Total time of flight
\(T=2t=\frac{2u}{g}=\sqrt{\frac{8H}{g}}\)
Velocity at height \(h\)
\(v_1=\sqrt{2g(H-h)}=u\sqrt{1-\frac{h}{H}}\)
Average speed for complete upward-downward trip
\(\bar v=\frac{u}{2}=\sqrt{\frac{gH}{2}}\)
Average velocity for complete trip
0
Distance covered in last second of upward motion
\(h_n=u-\frac{g}{2}(2n-1)=\frac{g}{2}\)
◉ Point Passed Twice:

Table 1: If body passes height h at times t₁ and t₂

Quantity
Formula
Total time of flight
\(T=t_1+t_2\)
Time to maximum height
\(t=\frac{t_1+t_2}{2}\)
Maximum height
\(H=\frac{g(t_1+t_2)^2}{8}\)
Initial velocity
\(u=\frac{g(t_1+t_2)}{2}\)
Height of that point
\(h=\frac{g}{2}t_1t_2\)
❖ Smooth Inclined Plane:
◉ Condition: Body released from height \(h\) along smooth inclined plane of inclination \(\theta\)

Table 1: Inclined Plane Formulae

Quantity
Formula
Velocity at bottom
\(v=\sqrt{2gh}=\sqrt{2gl\sin\theta}\)
Acceleration down plane
\(a=g\sin\theta\)
Time to slide down
\(t=\sqrt{\frac{2l}{g\sin\theta}}=\sqrt{\frac{2h}{g}}\csc\theta\)
Same height, different inclinations
\(t\propto \csc\theta\)
Equal length, different inclinations
\(t\propto \sqrt{\csc\theta}\)
◉ Ratios:
  • •
    Same height → \(t_1:t_2:t_3=\csc\theta_1:\csc\theta_2:\csc\theta_3\)
  • •
    Equal length → \(t_1:t_2:t_3=\sqrt{\csc\theta_1}:\sqrt{\csc\theta_2}:\sqrt{\csc\theta_3}\)
▢ Accelerates then Retards:
❖ Condition: Particle accelerates from rest with acceleration \(\alpha\) for time \(t_1\), distance \(x_1\), then retards to rest with retardation \(\beta\) for time \(t_2\), distance \(x_2\)

Table 1: Acceleration-Retardation Formulae

Quantity
Formula
Ratio
\(\frac{\alpha}{\beta}=\frac{t_2}{t_1}=\frac{x_2}{x_1}\)
Net acceleration
\(a=\frac{\alpha\beta}{\alpha+\beta}\)
Total time
\(t=t_1+t_2=\sqrt{\frac{2s}{a}}=\sqrt{\frac{2(x_1+x_2)(\alpha+\beta)}{\alpha\beta}}\)
Maximum velocity
\(v_{max}=at=\frac{\alpha\beta}{\alpha+\beta}(t_1+t_2)\)
Total distance
\(s=x_1+x_2=\frac{1}{2}at^2=\frac{1}{2}\frac{\alpha\beta}{\alpha+\beta}(t_1+t_2)^2\)
▢ Relative Velocity:
❖ Definition: Velocity of A with respect to B = velocity with which A appears to move when B is brought to rest
❖ Formula: \(\vec v_{AB}=\vec v_A-\vec v_B\)
❖ Crossing Formula: \(T=\frac{Sum\ of\ lengths}{Relative\ velocity}\)
❖ Train Problems:

Table 1: Crossing Time

Case
Formula
Train length \(l_1\) crossing bridge length \(l_2\)
\(T=\frac{l_1+l_2}{v}\)
Two trains moving opposite directions
\(T=\frac{l_1+l_2}{v_1+v_2}\)
Two trains moving same direction
\(T=\frac{l_1+l_2}{v_1-v_2}\)
❖ River Crossing:
  • •
    Least time
  • •
    Least distance
▢ Read and Digest:

Table 1: Special Formulae

Case
Formula / Result
A person throws \(n\) balls per second; second ball when first is at highest point
Height of each ball = \(\frac{g}{2n^2}\)
Two ends of accelerating train pass point with velocities \(u\) and \(v\)
Velocity of middle point = \(\sqrt{\frac{u^2+v^2}{2}}\)
Escalator: moving time \(t_1\), standing time \(t_2\)
Walking time = \(\frac{t_1t_2}{t_1+t_2}\)
Falling body stopped after \(t_1\), then released and reaches ground after \(t_2\)
\(t^2=t_1^2+t_2^2\)
Thrown upward from height \(h\) reaches ground in \(t_1\); thrown downward reaches in \(t_2\); released reaches in \(t\)
\(t=\sqrt{t_1t_2}\)
Body starts from rest with uniform acceleration
\(\frac{Distance\ in\ n^{th}\ second}{Distance\ in\ n\ seconds}=\frac{2}{n}-\frac{1}{n^2}\)
Train at speed \(v\) sights another train ahead moving same direction with speed \(v'\), separation \(x\)
Minimum retardation formula incomplete in given note
▢ High-Yield Recall:

Table 1: Motion in Straight Line One-Liners

Fact
Answer
Straight-line motion
Rectilinear motion
Distance
Scalar path length
Displacement
Vector shortest distance with direction
Distance-displacement relation
\(Distance \geq |Displacement|\)
Speed
Rate of change of distance
Velocity
Rate of change of displacement
Acceleration
Rate of change of velocity
Average speed
\(\frac{Total\ distance}{Total\ time}\)
Average velocity
\(\frac{Total\ displacement}{Total\ time}\)
Instantaneous speed
\(\frac{dx}{dt}\)
Instantaneous velocity
\(\frac{d\vec s}{dt}\)
Instantaneous acceleration
\(\frac{d\vec v}{dt}\)
Uniform motion
\(s=ut\)
First equation
\(v=u+at\)
Second equation
\(s=ut+\frac{1}{2}at^2\)
Third equation
\(v^2=u^2+2as\)
Nth second distance
\(s_{nth}=u+\frac{a}{2}(2n-1)\)
Successive distances from rest
\(1:3:5:...\)
Distances up to successive seconds from rest
\(1:4:9:...\)
Free fall time from height H
\(\sqrt{\frac{2H}{g}}\)
Impact velocity in free fall
\(\sqrt{2gH}\)
Maximum height upward
\(\frac{u^2}{2g}\)
Time to maximum height
\(\frac{u}{g}\)
Total time of flight upward-return
\(\frac{2u}{g}\)
Average velocity for complete vertical up-down motion
0
Acceleration on smooth inclined plane
\(g\sin\theta\)
Velocity at bottom of smooth inclined plane
\(\sqrt{2gh}\)
Same-height inclined plane time
\(t\propto \csc\theta\)
Equal-length inclined plane time
\(t\propto \sqrt{\csc\theta}\)
Relative velocity
\(\vec v_{AB}=\vec v_A-\vec v_B\)
Opposite trains crossing
\(T=\frac{l_1+l_2}{v_1+v_2}\)
Same-direction trains crossing
\(T=\frac{l_1+l_2}{v_1-v_2}\)
Equal displacement average velocity
Harmonic mean
Equal time average velocity
Arithmetic mean
Q1.
A train travels for 40 km with velocity of 80 km/hr and again it travels next 40 km with velocity of 40 km/hr. Then average speed of the train is:
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Q2.
A cyclist moving on a circular track of radius 40 m completes half revolution in 40 sec. Its average velocity is:
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Q3.
A person goes to office with 40 km/hr and returns home with 60 km/hr. What is his average speed?
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Q4.
A cyclist moving on a circular track of radius 40 m completes half a revolution in 40 s. Its average velocity is:
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Q5.
A car starts from rest and accelerates uniformly to 180 km/h in 10 s. The distance covered is:
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Q6.
When a body covers distance d₁ and d₂ at velocities v₁ and v₂ in equal time, and if v₁ is equal to v₂ then the average velocity is:
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Q7.
A lift of mass \(1000\ \mathrm{kg}\) accelerates vertically upward at \(1\ \mathrm{m/s^2}\). Find the tension in its supporting cable (take \(g=9.8\ \mathrm{m/s^2}\)).
📅BP 2013
Q8.
A boy started his journey from home to school which \(16\ \mathrm{km}\) far at uniform speed \(2.5\ \mathrm{km/h}\) and while returning, he returned with \(4\ \mathrm{km/h}\). What is his average speed?
📅BP 2012
Q9.
A person on a moving train throws upwards a coin, and if it falls behind him, then train is
📅BP 2012
Q10.
A boat moves at \(40\ \mathrm{m/s}\) perpendicular to a river flowing at \(30\ \mathrm{m/s}\). Find its resultant velocity magnitude relative to the ground.
📅BP 2012
Q11.
A lift of mass \(500\ \mathrm{kg}\) accelerates vertically upward at \(2\ \mathrm{m/s^2}\). Find the tension in its supporting cable (take \(g=9.8\ \mathrm{m/s^2}\)).
📅BP 2012
Q12.
A vehicle climbs a hill at speed of \(40\ \mathrm{km/h}\) and returns to same place at speed \(60\ \mathrm{km/h}\). What is the average speed for whole journey?
Q13.
A boat moves at \(4\ \mathrm{m/s}\) perpendicular to a river current. If its resultant speed is \(5\ \mathrm{m/s}\), find the current speed.
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Q14.
The slope of a velocity- time graph gives
📅MOE 2014
Q15.
Two bodies initially \(1\ \mathrm{m}\) apart vertically are released simultaneously from rest. Neglecting air resistance, what is their relative separation after \(3\ \mathrm{s}\) (before either hits the ground)?
📅MOE 2013
Q16.
A \(150\ \mathrm{m}\) train moving at \(6\ \mathrm{km/h}\) crosses a \(1.5\ \mathrm{km}\) bridge completely. How long does it take?
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Q17.
A body is thrown vertically upward and attains a velocity \(15\ \mathrm{m/s}\) at half the maximum height. The maximum height upto the body can reach will be:
📅MOE 2012•MOE 2011
Q18.
Body A moves north at \(3\ \mathrm{km/s}\), and B moves east at \(4\ \mathrm{km/s}\). Find the velocity of A relative to B.
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Q19.
A body moves \(30\ \mathrm{m}\) due north, \(20\ \mathrm{m}\) due east and \(30\sqrt{2}\) due south west. The total displacement covered by body from its initial position is:
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Q20.
Two bodies of mass '\(2\ \mathrm{m}\)' and 'm' are released from height '2H' and 'H' respectively. The ratio of time taken by\nthem to reach the ground is:
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Q21.
Two ships A and B are initially \(4\ \mathrm{km}\) apart, with A due west of B. Ship A moves south at \(6\ \mathrm{km/h}\), but B’s velocity is not specified. Find the magnitude of A’s velocity relative to B.
📅I.E 2013
Q22.
A body of mass \(200\ \mathrm{gm}\) is thrown upwards with initial velocity of \(30\ \mathrm{m/s}\). What is total energy of body at height of \(20\ \mathrm{m}\) from ground?
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Q23.
A body travels with velocity \(30\ \mathrm{m/s}\) for 1st half of time and with velocity \(40\ \mathrm{m/s}\) for 2nd half of time then what would be average velocity.
📅IOM 2013
Q24.
A body covers the first one-third of its journey at speed v₁ and the remaining two-thirds at speed v₂ in the same direction. Find its average velocity.
📅IOM 2012
Q25.
An \(80\ \mathrm{kg}\) person stands in a lift accelerating upward at \(4.9\ \mathrm{m/s^2}\). What does a weighing scale calibrated in kilogram-force read? (g = \(9.8\ \mathrm{m/s^2}\))
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Q26.
Two stones A and B are thrown from the top of a tower. The stone A is thrown vertically upward while the stone B is thrown vertically downward with the same speed. Which one of the following statements is true?
📅KU 2012
Q27.
A meter rod pivoted at its one end is rotated through \(120\ \mathrm{degree}\). Then displacement of its free end will be.
📅IOM
Q28.
A person walks four successive \(20\ \mathrm{m}\) straight segments, turning 90° left or right after each of the first three segments. Find the maximum net displacement.
Q29.
A person travels \(3\ \mathrm{km}\) towards north then \(2\ \mathrm{km}\) towards east and finally \(2 \sqrt{2}\) south-west. What is his displacement?
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Q30.
A flying kite travels \(24\ \mathrm{m}\) east then \(8\ \mathrm{m}\) south and finally \(6\ \mathrm{m}\) upward. What is its displacement from initial position?
Q31.
A racing car moves around a circular track of radius \(R\) at constant speed \(v\) and sweeps an angle \( heta\) (in radians). Find the magnitude of its average velocity over this interval.
Q32.
A car travelling along circular track at constant speed \(20\ \mathrm{m/s}\) has completed half revolution on the track. Its average velocity will be
📅IOM/MOE/KU
Q33.
The length of a second hand in a watch is \(1\ \mathrm{cm}\). The change in velocity in 15sec is
Q34.
If the displacement of a body is proportional to square of time. Then body has
📅IOM/I.E
Q35.
The displacement of a particle moving in a straight line at any instant time t is given as x = \(t^{2}\) + 20 (in metre). What is its average velocity for first four seconds?
Q36.
For x(t) = t² + \(20\ \mathrm{m}\), what is the instantaneous velocity at t = \(4\ \mathrm{s}\)?
Q37.
The acceleration of a particle at any instant of time 't' which starts from origin at initial velocity \(2\ \mathrm{m/s}\) is a = 2t (in m/\(s^{2}\)). What is its velocity in \(5\ \mathrm{seconds}\)?
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Q38.
A \(150\ \mathrm{m}\) train moves north at \(1\ \mathrm{m/s}\) while a parrot flies south parallel to the track at \(5\ \mathrm{m/s}\). How long does the parrot take to pass the entire train?
Q39.
Two boys start running towards each other from two points, they are \(120\ \mathrm{m}\) apart. One runs with a speed of \(5\ \mathrm{m/s}\) and other with a speed of \(7\ \mathrm{m/s}\). When and where do they meet each other from 1st point?
Q40.
A train of length \(200\ \mathrm{m}\) travelling at \(30\ \mathrm{m/s}\) overtakes another train of length \(300\ \mathrm{m}\) travelling at \(20\ \mathrm{m/s}\). The time taken by first train to pass the second is
Q41.
An insect crawls a distance of \(4\ \mathrm{m}\) along north in \(10\ \mathrm{seconds}\) and then a distance of \(3\ \mathrm{m}\) along east in \(5\ \mathrm{seconds}\). The average velocity of the insect is:
Q42.
A vehicle moving along a straight road covers half distance of its journey a \(40\ \mathrm{km/h}\) and next half distance at \(60\ \mathrm{km/h}\) in same direction. The average velocity for entire journey is
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Q43.
A vehicle moving along a straight road travels for half time at \(40\ \mathrm{km/h}\) and then travels for next half time at \(60\ \mathrm{km/h}\). The average velocity for entire journey is
📅IOM/MOE/I.E/KU/BPKIHS
Q44.
A body moving in a straight line travels \(2\ \mathrm{m/s}\) for first half distance and second half distance is covered in two equal time intervals at \(4\ \mathrm{m/s}\) and \(2\ \mathrm{m/s}\). What is its average velocity for entire journey?
Q45.
A body moving in a straight line travels \(2\ \mathrm{m/s}\) for first half time and for second half time it covers equal distance at velocities \(4\ \mathrm{m/s}\) and \(2\ \mathrm{m/s}\). What is its average velocity for entire journey?
Q46.
A person travels certain distance x due east at constant velocity v, and then he travels equal distance x due north at constant velocity v2. What is his average velocity for entire journey?
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Q47.
A person travels for certain time t due east at constant velocity v, and then he travels for equal time t due north at constant velocity v2. What is his average velocity for entire journey?
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Q48.
The displacement of a particle is given by x = 2 -5t + 6t^2 where x is in meters and t in seconds. The initial velocity of the particle
Q49.
If x denotes displacement in time t, and x = a cost, the acceleration is
Q50.
A car travelling due east at \(20\ \mathrm{m/s}\) turns towards north without changing speed in 10sec. The average acceleration of the car for its turn is
Q51.
A car moving east at \(20\ \mathrm{m/s}\) makes a U-turn, finishing at \(20\ \mathrm{m/s}\) west after \(10\ \mathrm{s}\). Find its average acceleration.
Q52.
A car travelling due north at \(30\ \mathrm{km/h}\) turns west and travels at the same speed, the change in velocity of car is
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Q53.
A body starts from rest, accelerates at constant rate a, then decelerates at rate b to rest after total time T. Find average velocity over the entire motion.
Q54.
A stone is dropped from the top of a tower. If it reaches the earth in \(6\ \mathrm{seconds}\), the height of the tower is nearly.
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Q55.
A car accelerates from rest at constant rate for the first 10seonds and covers a distance x. It covers a distance y in the next \(10\ \mathrm{seconds}\) at the same acceleration. Which of the velocity is true?
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Q56.
A car starts from rest, accelerates at a₁ for time t₁ and then decelerates at a₂ for time t₂ until it stops. Find t₁/t₂.
Q57.
A person throws two balls per second at regular intervals, throwing each new ball when the previous one reaches its greatest height. Find the greatest height (g = \(10\ \mathrm{m/s^2}\)).
Q58.
One ball is dropped from the top of a \(100\ \mathrm{m}\) tower. At the same instant, a second ball is thrown vertically upward from the ground at \(50\ \mathrm{m/s}\). When do they meet?
Q59.
A motorcycle starts and finishes at rest over a \(1.5\ \mathrm{km}\) journey. Its maximum acceleration is \(5\ \mathrm{m/s^2}\) and maximum braking deceleration is \(10\ \mathrm{m/s^2}\). Find the minimum travel time.
Q60.
Two balls A and B are simultaneously projected from the top of a building at \(10\ \mathrm{m/s}\) upwards and \(20\ \mathrm{m/s}\) downwards respectively. Find out the separation between them 3sec after projection.
Q61.
A stone would fall from a height in \(5\ \mathrm{s}\) if uninterrupted. Instead it is stopped and released again from rest after the first \(3\ \mathrm{s}\) of falling. Ignoring stopping duration, find total falling time.
Q62.
A balloon is going upwards with velocity \(12\ \mathrm{m/s}\). It releases a packet when it is a height of \(65\ \mathrm{m}\) from ground. How much time will the packet take to reach the ground (g = \(10\ \mathrm{m/s^2}\))?
Q63.
A car moving with a speed of \(40\ \mathrm{km/h}\) can be stopped by applying brakes after at least \(2\ \mathrm{m}\). If the same car is moving with speed of \(80\ \mathrm{km/h}\), the minimum stopping distance is
Q64.
A ball is dropped from the top of a very high tower. Distance covered by it in last second of its motion equal to the distance covered by in first 3secs of its motion. Find the time of fall
Q65.
A ball moving vertically upward travels equal distances during the 4th and 5th seconds after projection. Find its initial speed (g = \(10\ \mathrm{m/s^2}\)).
Q66.
A ball is projected vertically upwards from the ground. It is found at the same elevation at t = \(3\ \mathrm{s}\) and t = \(7\ \mathrm{s}\) after projection. Find the projection speed.
Q67.
A stone is dropped from the top of a tower. If it covers \(24.5\ \mathrm{m}\) in the last second of its motion, the height of the tower is
Q68.
A body is projected vertically upward from point A, the top of a tower. It reaches the ground in t1 secs. If it is projected vertically downwards from A with the same velocity, it reaches the ground in t2 secs. If it falls freely from A, it would reach the ground in
Q69.
A body is released from top of a smooth inclined plane having inclination \(30\ \mathrm{degree}\) and takes 3secs to reach the bottom. If the angle of inclination is doubled keeping the height same, what will be the time taken for the same process?
Q70.
A body sliding on a smooth inclined plane requires \(4\ \mathrm{seconds}\) to reach the bottom, starting from rest at the top. How much time does it take to cover one-fourth distance starting from rest at the top?
📅KU
Q71.
A ball is dropped from the top of a building and its time of fall is 't'. In the last 1/4^th time of its fall, it travels 1.4 metres. Find out the height of the building
Q72.
A ball is projected vertically upwards from ground and its time of rise is 't'. In first t/5 time, it covers a distance of \(2.7\ \mathrm{m}\). Find the height
Q73.
A body is thrown upwards with velocity \(100\ \mathrm{m/s}\) and it travels \(5\ \mathrm{m}\) in the last second of its upward journey. If the same body is thrown upwards with velocity \(200\ \mathrm{m/s}\), what distance will it travel in the last second of upward journey?
Q74.
A body falls freely from rest and has velocity v after it falls through a height h. The distance it has to fall down further for its velocity to become double is
Q75.
A ball dropped downwards. After \(1\ \mathrm{second}\) another ball is dropped downwards from the same point. What is the distance between them \(3\ \mathrm{seconds}\) after the first ball was dropped?
Q76.
Two bodies are thrown vertically upward with their initial velocities in ratio 2: 3. Then the ratio of maximum heights attained by them is
Q77.
Ball A is projected vertically upward with initial speed u while ball B is simultaneously released from rest. Find the velocity of A relative to B after time t, before either hits the ground.
Q78.
A stone dropped from a height covers 5/9 part of total distance in last second. Then initial height will be
Q79.
Water drops are falling at regular intervals of time from a roof \(5\ \mathrm{m}\) high. When first drop strikes the ground, third drop just leaves the roof then height of the second drop from the ground at this\ninstant is
Q80.
A body thrown vertically upwards attains a maximum height H. While moving upwards if it covers first (3H/4) distance from the ground in time T, then time taken to cover remaining distance H/4 will be
📅IOM/MOE/KU/BPKIHS
Q81.
A ball rises from ground to its highest point in \(10\ \mathrm{s}\) under ordinary air resistance, then falls back to the same level. Compared with \(10\ \mathrm{s}\), the fall time is generally:
Q82.
A bullet loses 1/20 of its velocity after penetrating a plank. How may planks are required to stop the bullet?
Q83.
Three particles A, B and C are situated at the vertices of an equilateral triangle ABC of side l at t = 0. Each of the particle moves with constant speed v. A always has its velocity along AB, B along BC and C along CA. At what time will these particles meet each other?
Q84.
A vehicle moving at constant acceleration along a straight road has velocities u and v at two point A and B on the road. Then it velocity at midway between A and B will be
Q85.
A person who can swim at \(5\ \mathrm{km/h}\) in still water, crosses a river \(1\ \mathrm{km}\) wide flowing at \(3\ \mathrm{km/h}\) along shortest route. Then time taken to cross the river is
Q86.
A person who can swim at \(12\ \mathrm{km/h}\) in still water wants to cross a river flowing at \(6\ \mathrm{km/h}\) along shortest route. Then he has to start swimming at
Q87.
A person who can swim at \(5\ \mathrm{km/h}\) in still water cross a river \(1\ \mathrm{km}\) wide flowing at \(3\ \mathrm{km/h}\) in shortest time. How far he will reach at another bank?
Q88.
To a person running east at \(8\ \mathrm{km/h}\), rain appears to fall vertically at \(6\ \mathrm{km/h}\). What is the actual velocity of the rain?
Q89.
A boat takes 2 h to travel \(8\ \mathrm{km}\) outward and \(8\ \mathrm{km}\) back in still water. If river current speed is \(4\ \mathrm{km/h}\), find the upstream-and-return time for the same distances.
Q90.
A stone is dropped from the top of a tower of height 'h'. It reaches the ground in 't' secs. The position of the stone after t/3 secs. will be ... from the ground
📅IOM 063
Q91.
A stone is dropped from the top of tower of height h. After \(1\ \mathrm{second}\) another stone is dropped from balcony \(20\ \mathrm{m}\) below the top. Both reach the bottom simultaneously. What is the value of h?\n
📅IOM 05
Q92.
A body of mass 'm' is released from height 'h' in time 't'. Then, acceleration is determined by:\n
📅IOM 2015
Q93.
When an aeroplane is moving with velocity \(600\ \mathrm{km/h}\) due east & return with \(400\ \mathrm{km/h}\),\nthen what is average speed if they travel same distance?
📅KU 2016
Q94.
A person is traveling at \(4\ \mathrm{m/s}\) towards east. The rain is apparently falling vertically downwards with \(3\ \mathrm{m/s}\), then the actual velocity of rain is,
📅IOM 2016
Q95.
A train travels for \(40\ \mathrm{km}\) with velocity of \(80\ \mathrm{km/h}\) and again it travels next \(40\ \mathrm{km}\) with velocity of \(40\ \mathrm{km/h}\). Then average speed of the train is
📅IOM 2016