7Work, Energy, Power and Collision

📚
WORK, ENERGY, POWER AND COLLISION
▢ Work:
❖ Definition: Work is done when a body undergoes displacement under action of force
❖ Zero Work Examples:
  • •
    Man pushes object but displacement = 0
  • •
    Man rows boat upstream but remains at rest w.r.t. shore
  • •
    Centripetal force in circular motion
  • •
    Gravitational pull on satellite in circular orbit
  • •
    Person carrying load on head on horizontal road
  • •
    Car moving with uniform speed on frictionless road
❖ Formulae:

Table 1: Work Done

Case
Formula
Constant force
\(W=\vec F\cdot\vec s=Fs\cos\theta\)
Variable force
\(W=\int _{s_1}^{s_2}\vec F\cdot d\vec s=\int _{s_1}^{s_2}F\cos\theta\ ds\)
Work-energy theorem
\(W=\Delta K.E.\)
SI unit
Joule
CGS unit
Erg
Conversion
\(1\ Joule=10^7\ erg\)
❖ Sign of Work:

Table 1: Positive, Negative and Zero Work

Angle \(\theta\)
Work
Example
\(0\leq\theta<\frac{\pi}{2}\)
Positive
Force supports displacement
\(\frac{\pi}{2}<\theta\leq\pi\)
Negative
Friction; gravity during upward motion
\(\theta=\frac{\pi}{2}\)
Zero
Centripetal force
❖ Important Points:
  • •
    Work is scalar
  • •
    Work done does not depend on time taken
  • •
    Work done by a body → energy decreases
  • •
    Work done on a body → energy increases
▢ Power:
❖ Definition: Rate of doing work

Table 1: Power Formulae

Quantity
Formula / Unit
Power
\(P=\frac{Work}{Time}\)
Using force and displacement
\(P=\frac{\vec F\cdot\vec s}{t}\)
Instantaneous power
\(P=\vec F\cdot\vec v=Fv\cos\theta\)
Rate of energy change
\(P=\frac{\Delta E}{t}\)
SI unit
Watt
1 Watt
\(1\ W=1\ J\ s^{-1}\)
Practical unit
Horse power / HP
1 HP
\(746\ W\)
▢ Energy:
❖ Definition: Capacity of a body to do work
❖ Nature:
  • •
    Scalar quantity
  • •
    Same units and dimensions as work
  • •
    Mechanical energy = kinetic energy + potential energy
❖ Kinetic Energy:
◉ Definition: Energy possessed by body due to motion

Table 1: Kinetic Energy Formulae

Quantity
Formula / Point
Kinetic energy
\(E_k=\frac{1}{2}mv^2\)
In terms of momentum
\(E_k=\frac{p^2}{2m}\)
Momentum
\(p=\sqrt{2mE_k}\)
If \(E_k=0\)
\(p=0\)
Frame dependence
K.E. depends on frame of reference
◉ Frame Example: Person sitting in train: K.E. = 0 in train frame; K.E. = \(\frac{1}{2}mv^2\) in earth frame
◉ Stopping by Same Force:

Table 1: Stopping Distance and Time

Quantity
Formula
Stopping distance ratio
\(\frac{s_1}{s_2}=\frac{m_1v_1^2}{m_2v_2^2}\)
Stopping time ratio
\(\frac{t_1}{t_2}=\frac{m_1v_1}{m_2v_2}\)
◉ Bullet Through Planks:

Table 1: Bullet Special Cases

Condition
Result
Bullet loses \(\frac{1}{n}\) of velocity per plank
No. of planks = \(\frac{n}{2}+0.5\) if \(n\) odd; \(\frac{n}{2}+1\) if \(n\) even
Bullet loses \(\frac{1}{n}\) of velocity after penetrating distance \(x\)
Total distance before rest = \(\frac{n^2}{2n-1}x\)
Further distance after penetrating \(x\)
\(\frac{(n-1)^2}{2n-1}x\)
Bullet loses \(\frac{1}{n}\) of K.E. per plank
No. of planks = \(n\)
Bullet loses \(\frac{1}{n}\) of K.E. after distance \(x\)
Total distance = \(nx\)
Further distance after distance \(x\)
\((n-1)x\)
❖ Potential Energy:
◉ Definition: Energy stored due to configuration or position in a field
◉ Gravitational PE:

Table 1: Gravitational Potential Energy

Quantity
Formula
Potential energy at height \(h\)
\(U=-\frac{GMm}{R+h}=-\frac{mgR^2}{R+h}\)
Change in PE from surface to height \(h\)
\(\Delta U=\frac{mghR}{R+h}\)
If \(h\ll R\)
\(\Delta U=mgh\)
If \(h\gg R\)
\(\Delta U=mgR\)
◉ Special PE Formulae:

Table 1: Potential Energy Special Cases

Case
Increase in PE
Stick of length \(l\), pivoted at one end, displaced by \(\theta\)
\(\Delta U=\frac{mgl}{2}(1-\cos\theta)\)
Pendulum bob displaced by angle \(\theta\)
\(\Delta U=mgl(1-\cos\theta)\)
Elastic potential energy
\(U=\frac{1}{2}kx^2\)
◉ Sign of PE:
  • •
    Attractive forces → PE negative
  • •
    Repulsive forces → PE positive
❖ Work-Energy Theorem:
◉ Statement: Work done on a body equals change in kinetic energy
◉ Formula: \(W=\Delta K.E.=\frac{1}{2}mv_2^2-\frac{1}{2}mv_1^2\)
◉ Applications:

Table 1: Work-Energy Applications

Case
Work done / Result
Empty full tank to half depth
\(W=\frac{3}{8}Mgh\)
Take out all liquid from full tank
\(W=\frac{Mgh}{2}\)
Chain length \(l\), \(\frac{l}{n}\) hanging
\(W=\frac{mgl}{2n^2}\)
Released chain: velocity when end leaves edge
\(v=\frac{1}{n}\sqrt{(n^2-1)gl}\)
Rod of mass \(m\), length \(l\), made to stand on one end
\(W=\frac{mgl}{2}\)
❖ Conservative and Non-conservative Forces:

Table 1: Conservative vs Non-conservative Forces

Feature
Conservative force
Non-conservative force
Mechanical energy
Conserved
Not conserved
Work done
Depends only on initial and final positions
Depends on actual path
Closed path work
Zero
Not zero
Examples
Gravitational, electrostatic, magnetostatic, all central forces
Frictional, viscous forces
▢ Collision:
❖ Definition: Mutual interaction between particles for short time interval in which momentum and kinetic energy may change
❖ Law of Collision:
◉ Statement: Velocity of separation after collision is directly proportional to velocity of approach before collision

Table 1: Coefficient of Restitution

Quantity
Formula / Point
Velocity of separation
\(v_2-v_1\)
Velocity of approach
\(u_1-u_2\)
Relation
\(v_2-v_1=e(u_1-u_2)\)
Coefficient of restitution
\(e=\frac{v_2-v_1}{u_1-u_2}\)
Nature
Dimensionless and unitless
Elastic collision
\(e=1\)
Perfectly inelastic collision
\(e=0\)
Practical collision
\(0
◉ Notes:
  • •
    Linear momentum is conserved in every collision
  • •
    Total energy is conserved in every collision
  • •
    Kinetic energy is conserved only in elastic collision
❖ Types of Collision:

Table 1: Elastic vs Inelastic Collision

Feature
Elastic collision
Inelastic collision
Total energy
Conserved
Conserved
Linear momentum
Conserved
Conserved
Kinetic energy
Conserved
Not conserved
Coefficient of restitution
\(e=1\)
\(e<1\)
Perfectly inelastic collision
—
\(e=0\)
❖ Elastic Collision in One Dimension:
◉ Momentum Conservation: \(m_1u_1+m_2u_2=m_1v_1+m_2v_2\)
◉ Kinetic Energy Conservation: \(\frac{1}{2}m_1u_1^2+\frac{1}{2}m_2u_2^2=\frac{1}{2}m_1v_1^2+\frac{1}{2}m_2v_2^2\)
◉ Final Velocities:

Table 1: Elastic Collision Final Velocities

Quantity
Formula
\(v_1\)
\(v_1=\frac{(m_1-m_2)u_1+2m_2u_2}{m_1+m_2}\)
\(v_2\)
\(v_2=\frac{2m_1u_1+(m_2-m_1)u_2}{m_1+m_2}\)
◉ Special Cases:

Table 1: Elastic Collision Special Cases

Condition
Result
\(u_2=0\)
\(v_1=\frac{m_1-m_2}{m_1+m_2}u_1\), \(v_2=\frac{2m_1}{m_1+m_2}u_1\)
\(m_1=m_2\)
\(v_1=u_2\), \(v_2=u_1\); velocities exchange
\(m_1=m_2\), \(u_2=0\)
\(v_1=0\), \(v_2=u_1\)
\(m_1\ll m_2\), \(u_2=0\)
\(v_1\approx -u_1\), \(v_2\approx 0\)
\(m_1\gg m_2\), \(u_2=0\)
\(v_1\approx u_1\), \(v_2\approx 2u_1\)
❖ Inelastic Collision:
◉ Final Velocities with Coefficient e:

Table 1: Inelastic Collision Final Velocities

Quantity
Formula
\(v_1\)
\(v_1=\frac{m_1u_1+m_2u_2+em_2(u_2-u_1)}{m_1+m_2}\)
\(v_2\)
\(v_2=\frac{m_1u_1+m_2u_2+em_1(u_1-u_2)}{m_1+m_2}\)
◉ Perfectly Inelastic Collision:

Table 1: Perfectly Inelastic Collision

Quantity
Formula / Point
Bodies move with common velocity
\(v\)
Momentum equation
\(m_1u_1+m_2u_2=(m_1+m_2)v\)
Common velocity
\(v=\frac{m_1u_1+m_2u_2}{m_1+m_2}\)
Coefficient of restitution
\(e=0\)
Loss in K.E.
\(\frac{1}{2}\frac{m_1m_2}{m_1+m_2}(u_1-u_2)^2\)
❖ Body Dropped and Rebounds:
◉ Condition: Body dropped from height \(h_0\), strikes ground and rebounds to height \(h_1\)

Table 1: Rebound Formulae

Quantity
Formula
Striking velocity
\(u=\sqrt{2gh_0}\)
Velocity of bounce
\(v=eu=e\sqrt{2gh_0}\)
Coefficient of restitution
\(e=\frac{v}{u}=\sqrt{\frac{h_1}{h_0}}\)
Height after \(n\) bounces
\(h_n=e^{2n}h_0\)
Velocity after \(n\) bounces
\(v_n=e^n u\)
Momentum after \(n\) bounces
\(p_n=e^n p_0\)
Fractional loss in momentum after \(n\) bounces
\(\frac{\Delta p}{p_0}=1-e^n\)
K.E. after \(n\) bounces
\(K.E._n=e^{2n}K.E._0\)
Fractional loss in K.E. after \(n\) bounces
\(\frac{\Delta K.E.}{K.E._0}=1-e^{2n}\)
❖ Oblique Elastic Collision:
◉ Condition: Body of mass \(m_1\) collides elastically with body of mass \(m_2\), then both move at angles \(\alpha\), \(\beta\)

Table 1: Oblique Collision Equations

Conservation
Equation
Kinetic energy
\(\frac{1}{2}m_1u_1^2+\frac{1}{2}m_2u_2^2=\frac{1}{2}m_1v_1^2+\frac{1}{2}m_2v_2^2\)
Momentum along X-axis
\(m_1u_1+m_2u_2=m_1v_1\cos\alpha+m_2v_2\cos\beta\)
Momentum along Y-axis
\(m_1v_1\sin\alpha-m_2v_2\sin\beta=0\)
◉ Special Case: If \(m_1=m_2\) and \(u_2=0\), then \(\alpha+\beta=90^\circ\)
▢ Machines:

Table 1: Simple Machine Terms

Term
Formula
Mechanical advantage / MA
\(MA=\frac{Load}{Effort}=\frac{W}{P}\)
Velocity ratio / VR
\(VR=\frac{Distance\ travelled\ by\ effort}{Distance\ travelled\ by\ load}=\frac{D}{d}\)
Efficiency
\(\eta=\frac{MA}{VR}=\frac{W/P}{D/d}<1\)
Relation
\(MA
▢ Read and Digest:

Table 1: Important Points

Fact
Point / Formula
Flowing water
Has both K.E. and P.E.
Potential energy
Exists only for conservative forces
Energy change
Work is necessarily done
Static/dynamic equilibrium
Net work done = 0
Unstable equilibrium
Potential energy maximum
Air bubble rises in water
P.E. decreases because work is done by upthrust
Machine gun fires \(n\) bullets/s, each K.E. \(K\)
\(P=nK\)
Force to hold machine gun
\(F=np=n\sqrt{2mK}\)
Oscillation
Occurs about stable equilibrium only
Slope of work-time graph
\(P=\frac{dW}{dt}=\tan\theta\)
Gravity work
Positive downward, negative upward
Constant power motion
\(v\propto t^{1/2}\), \(s\propto t^{3/2}\), \(a\propto t^{-1/2}\)
Hydrogen balloon in air
Weight is negative; gravitational P.E. decreases with altitude
Like charges brought together
P.E. increases
Unlike charges brought together
P.E. decreases
Centripetal force
Power dissipated = 0
Water through pipe speed \(v\)
\(P=\frac{1}{2}\rho Av^3\)
Mass \(m\) from rest under constant force \(F\) for time \(t\)
Maximum power = \(\frac{F^2t}{m}\)
▢ High-Yield Recall:

Table 1: Work, Energy, Power and Collision One-Liners

Fact
Answer
Work
\(W=\vec F\cdot\vec s=Fs\cos\theta\)
Work nature
Scalar
SI unit of work
Joule
CGS unit of work
Erg
Joule-erg relation
\(1J=10^7\ erg\)
Work by centripetal force
Zero
Work by friction
Negative
Power
\(P=\frac{W}{t}=\vec F\cdot\vec v\)
1 HP
\(746\ W\)
Energy
Capacity to do work
Kinetic energy
\(\frac{1}{2}mv^2\)
K.E. in terms of momentum
\(\frac{p^2}{2m}\)
Momentum in terms of K.E.
\(p=\sqrt{2mE_k}\)
Elastic P.E.
\(\frac{1}{2}kx^2\)
Work-energy theorem
\(W=\Delta K.E.\)
Conservative force closed path work
Zero
Conservative forces
Gravitational, electrostatic, magnetostatic, central forces
Non-conservative forces
Frictional, viscous
Collision
Short-time mutual interaction
Coefficient of restitution
\(e=\frac{v_2-v_1}{u_1-u_2}\)
Elastic collision
\(e=1\)
Perfectly inelastic collision
\(e=0\)
Conserved in every collision
Linear momentum and total energy
K.E. conserved only in
Elastic collision
Equal masses elastic collision
Velocities exchange
Perfectly inelastic common velocity
\(v=\frac{m_1u_1+m_2u_2}{m_1+m_2}\)
Loss in K.E. in perfectly inelastic collision
\(\frac{1}{2}\frac{m_1m_2}{m_1+m_2}(u_1-u_2)^2\)
Rebound coefficient
\(e=\sqrt{\frac{h_1}{h_0}}\)
Height after n bounces
\(h_n=e^{2n}h_0\)
Velocity after n bounces
\(v_n=e^n u\)
Oblique elastic equal masses
Move perpendicular after collision
Mechanical advantage
\(MA=\frac{W}{P}\)
Velocity ratio
\(VR=\frac{D}{d}\)
Efficiency
\(\eta=\frac{MA}{VR}\)
Q1.
Two bodies of different masses possess equal kinetic energy. Their linear momentum will be
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Q2.
Ship of mass 3×10^8 kg is pulled by 6×10^5 N through 3 m. Speed is
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Q3.
A car can be stopped over a distance x when its momentum is p. What will be the stopping distance when momentum is 2p?
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Q4.
If two bodies of masses 4M and 16M move with same kinetic energy, what is the ratio of their momentum?
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Q5.
'A' and 'B' are two bodies and A is lighter than B. If they have equal momentum, then the kinetic energy of A is:
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Q6.
If the product of mass and velocity is doubled, the new kinetic energy is how many times the original kinetic energy?
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Q7.
When momentum of a body is increased by 100%, its kinetic energy increases by:
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Q8.
A body of mass \(m\) moving with velocity \(u\) collides with another body of the same mass at rest and they become embedded. The loss of kinetic energy is:
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Q9.
A train of mass 'm' is moving with velocity 'v' making an angle '6' with horizontal straight line power delivered is?
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Q10.
If kinetic energy changes by 4%, the momentum changes by approximately:
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Q11.
If a stone of mass 'M' dropped from height 'h' penetrates distance 's' into sand, the average resistance offered by the sand to the stone is:
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Q12.
What happens to the linear momentum of the projectile when its velocity is doubled?
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Q13.
A car is going with a linear momentum p. When brakes are applied, it comes to a stop in a distance s. If the same car were going with a linear momentum 2p and the brakes are applied (assume that the brake force is same in the two cases), it comes to a stop in a distance of:
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Q14.
For a constant power, the velocity of a body depends on displacement as:
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Q15.
A bullet of mass 10 g moving with a velocity 200 m/s hits a wooden block and penetrates 2 cm before coming to rest. What is its average force exerted on bullet during penetration?
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Q16.
If an object fall from top of inclined plane and get velocity 'u' in ground. How much distance will it cover if it is projected upward with that velocity?
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Q17.
If a lift of mass \(1000 \mathrm{kg}\) moves upward with acceleration of \(1 \mathrm{m/s}\) then tension is:
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Q18.
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?
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Q19.
A person on a moving train throws upwards a coin, and if it falls behind him, then train is
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Q20.
A boat was crossing the river with velocity \(40 \mathrm{m/s}\) and river flowing with velocity \(30 \mathrm{m/s}\). Then resultant velocity was:
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Q21.
If a lift of mass \(500 \mathrm{Kg}\) moves upward with acceleration \(2 \mathrm{m/s}\) then the tension is:
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Q22.
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?
Q23.
A man intends to cross a river on a boat with velocity \(4 \mathrm{m/s}\). If resultant velocity of boat is \(5 \mathrm{m/s}\). Then velocity of river is:
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Q24.
The slope of a velocity- time graph gives
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Q25.
Two bodies, one held I'm above the other directly, are released simultaneously and fall freely under gravity. After \(3 second\) their relative separation will be
📅MOE 2013
Q26.
Time taken by a train of length \(150 \mathrm{m}\) and traveling with a uniform velocity of \(6 \mathrm{km/h}\) to cross completely a bridge of length \(1.5 \mathrm{km}\) will be:
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Q27.
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:
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Q28.
A body A is moving in north with \(3 \mathrm{km/s}\) and B with \(4 \mathrm{km/s}\) east. What is the relative velocity of a body A with respect to B.
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Q29.
A body moves \(30m\) due north, \(20m\) due east and \(30\sqrt{2}\) due south west. The total displacement covered by body from its initial position is:
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Q30.
Two bodies of mass '\(2 \mathrm{m}\)' and 'm' are released from height '2H' and 'H' respectively. The ratio of time taken by them to reach the ground is:
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Q31.
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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Q32.
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.
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Q33.
A body cover 1/3 rd distance with V1, velocity next 2/3 rd distance with V2 velocity the average velocity.
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Q34.
A person weighting \(80 \mathrm{kg}\) is standing on lift moves upward with a uniform acceleration of \(4.9 \mathrm{m/s^2}\) then apparent wt. of the person is:
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Q35.
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?
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Q36.
A meter rod pivoted at its one end is rotated through \(120^{\circ}\). Then displacement of its free end will be.
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Q37.
A person turns left or right by 90 after travelling each \(20m\) in straight line. What is his maximum displacement after three successive turns?
Q38.
A person travels \(3km\) towards north then \(2km\) towards east and finally \(2 \sqrt{2}\) south-west. What is his displacement?
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Q39.
A flying kite travels \(24m\) east then \(8m\) south and finally \(6m\) upward. What is its displacement from initial position?
Q40.
A racing car moving along circular track of radius R at constant speed v has described angle theta about centre of the track n certain time. What is the average velocity for the interval of time?
Q41.
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]
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Q42.
The length of a second hand in a watch is \(1cm\). The change in velocity in \(15sec\) is
Q43.
If the displacement of a body is proportional to square of time. Then body has
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Q44.
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?
Q45.
In the above question, what is its velocity at \(4 second\)?
Q46.
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 \(5 seconds\)?
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Q47.
A train of 150 meter length is going towards north direction at a speed of \(1 \mathrm{m/s}\). A parrot flies at the speed of \(5\mathrm{m/s}\) towards south direction parallel to the railway track. The time taken by the parrot to cross the train is
Q48.
Two boys start running towards each other from two points, they are \(120m\) 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?
Q49.
A train of length \(200m\) travelling at \(30\mathrm{m/s}\) overtakes another train of length \(300m\) travelling at \(20\mathrm{m/s}\). The time taken by first train to pass the second is
Q50.
An insect crawls a distance of \(4m\) along north in \(10seconds\) and then a distance of \(3m\) along east in \(5 seconds\). The average velocity of the insect is:
Q51.
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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Q52.
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
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Q53.
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?
Q54.
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?
Q55.
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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Q56.
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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Q57.
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
Q58.
If x denotes displacement in time t, and \(x = a cost\), the acceleration is
Q59.
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
Q60.
In the above question if the car makes 'U' turn then average acceleration of the car will be
Q61.
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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Q62.
A body starts from rest and moves at constant acceleration a in a straight line for time t1, covering a distance x1, and then it retards at rest at constant deceleration b for time t2 covering a distance X2. Then average velocity will be
Q63.
A stone is dropped from the top of a tower. If it reaches the earth in \(6 seconds\), the height of the tower is nearly.
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Q64.
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 \(10 seconds\) at the same acceleration. Which of the velocity is true?
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Q65.
A car accelerates at a constant rate a1 for time t1 and then retards at the constant rate a2 for time t2 and comes to rest \(t_{1}/t_{2}\) =
Q66.
A person is throwing balls into the air one after the other. He throws the second ball when first ball is at the highest point. If he is throwing two balls every second, how high do they rise?
Q67.
A ball is dropped from the top of a tower \(100m\) high. Simultaneously, another ball is thrown upwards with a speed of \(50\mathrm{m/s}\). After what time do they cross each other?
Q68.
The engine of a motorcycle can produce a maximum acceleration \(5\mathrm{m/s}\). Its brakes can produce a maximum retardation \(10 \mathrm{m/s^2}\). What is the minimum time in which it can cover a distance of \(1.5 \mathrm{km}\)?
Q69.
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.
Q70.
59.A stone is dropped from a certain height which can reach the ground is \(5 second\). It is stopped after three second of its fall and then is again released. The total time taken by the stone to reach the ground will be
Q71.
A balloon is going upwards with velocity \(12\mathrm{m/s}\). It releases a packet when it is a height of \(65m\) from ground. How much time will the packet take to reach the ground
Q72.
A car moving with a speed of \(40\mathrm{km/h}\) can be stopped by applying brakes after at least \(2m\). If the same car is moving with speed of \(80 \mathrm{km/h}\), the minimum stopping distance is
Q73.
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
Q74.
A ball thrown vertically upward covers equal distances in its fourth and fifth second. Then initial velocity of projection will be
Q75.
A ball is projected vertically upwards from the ground. It is found at the same elevation at t = \(3s\) and t = \(7s\) after projection. Find the projection speed.
Q76.
A stone is dropped from the top of a tower. If it covers \(24.5m\) in the last second of its motion, the height of the tower is
Q77.
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
Q78.
A body is released from top of a smooth inclined plane having inclination \(30^{\circ}\) 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?
Q79.
A body sliding on a smooth inclined plane requires \(4 seconds\) 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
Q80.
A ball is dropped from the top of a building and its time of fall is 't'. In the last \(\frac{1}{4}^th\) time of its fall, it travels \(1.4 metres\). Find out the height of the building
Q81.
A ball is projected vertically upwards from ground and its time of rise is 't'. In first \(\frac{t}{5}\) time, it covers a distance of \(2.7m\). Find the height
Q82.
A body is thrown upwards with velocity \(100\mathrm{m/s}\) and it travels \(5m\) 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?
Q83.
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
Q84.
A ball dropped downwards. After \(1 second\) another ball is dropped downwards from the same point. What is the distance between them \(3seconds\) after the first ball was dropped?
Q85.
Two bodies are thrown vertically upward with their initial velocities in ratio 2: 3. Then the ratio of maximum heights attained by them is
Q86.
A ball A is thrown vertically upward at initial speed u while another B is dropped from a height at the same instant. After time t, their relative velocity w.r.t one another will be
Q87.
A stone dropped from a height covers 5/9 part of total distance in last second. Then initial height will be
Q88.
Water drops are falling at regular intervals of time from a roof \(5m\) high. When first drop strikes the ground, third drop just leaves the roof then height of the second drop from the ground at this instant is
Q89.
A body thrown vertically upwards attains a maximum height H. While moving upwards if it covers first (\(\frac{3H}{4}\)) distance from the ground in time T, then time taken to cover remaining distance \(\frac{H}{4}\) will be
📅IOM•MOE•KU•BPKIHS
Q90.
When a ball is thrown upwards with air resistance not neglected, it takes \(10seconds\) to reach a height. It then returns to ground in time.
📅MOE
Q91.
A bullet loses 1/20 of its velocity after penetrating a plank. How may planks are required to stop the bullet?
Q92.
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?
Q93.
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
Q94.
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
Q95.
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
Q96.
A person who can swim at \(5\mathrm{km/h}\) in still water cross a river \(1km\) wide flowing at \(3\mathrm{km/h}\) in shortest time. How far he will reach at another bank?
Q97.
To a person running due east at \(8\mathrm{km/h}\) raindrops appear to fall vertically downward at \(6\mathrm{km/h}\). Then actual velocity of raindrops is
Q98.
A boat takes \(2 hours\) to travel \(8km\) and back in still water lake. If the velocity of water is \(4\mathrm{km/h}\), the time taken for going upstream of \(8km\) and coming back is
Q99.
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 \(\frac{t}{3}\) secs. will be ... from the ground
📅IOM 063
Q100.
A stone is dropped from the top of tower of height h. After \(1 second\) another stone is dropped from balcony \(20 \mathrm{m}\) below the top. Both reach the bottom simultaneously. What is the value of h?
📅IOM 05
Q101.
A body of mass 'm' is released from height 'h' in time 't'. Then, acceleration is determined by:
📅IOM 2015
Q102.
When an aeroplane is moving with velocity \(600\mathrm{km/h}\) due east & return with \(400\mathrm{km/h}\), then what is average speed if they travel same distance?
📅KU 2016
Q103.
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
Q104.
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