For motion on curved path, distance/displacement > 1
For motion on straight path, distance/displacement = 1
For falling body, distance/displacement = 1
For motion on curved path, speed/velocity > 1
For motion on straight path, speed/velocity = 1
An acclration is said to be uniform if a. Average accelration = instantaneous accelration b. Velocity changes uniformly
When velocity is constant, then velocity time graph is a horizontal straight line
If accelration is constant, then velocity time graph is a straight line which rises same height for equal intervals of time
If acclration is variable, then velocity time graph is curved
Suppose velocity of a body starts from 0 to v in time t then average velocity will be Vav = 0 + v/2 = v/2
Equations for linear motion with uniform accelration are following a. vf = vi + at b. S = Vav × t = (vf + vi/2) × t c. S = vit + 1/2 g² d. vf² – vi² = 2aS e. S = vt = v × underroot 2H/g
For vertical upward motion, use negative sign with accelration in these equations. Example: A body is thrown vertically upward with velocity 80m/s. How much height it will cover in 10sec? H= vit-0.5gt² = 80×10-0.5(10)(10)² = 800-500=300m/s
To find distance covered in nth sec S = 5(2tn – 1) Example: A body is fallen from certain height. How much distance it will cover in 4th sec? S = 5(2×4 – 1) = 35m
To find distance covered in n sec S = 5t² Example: A body is fallen from certain height. How much distance it will cover in 4 sec? S = 5(4)² = 80m
Two bodies of same masses moving with different velocities are stopped by same force. The relation between their velocities and distances in which they are stopped is FS = mv²/2 S = v² S1/S2 = v1²/v2²
Newton’s Law of motion are applicable for only low speeds as compared to speed of light
For fast moving objects, relativistic mechanics developed by Albert Einstein. It is applicable for atomic particles
Newton ist law is also called law of inertia.
Inertia of a body is directly proportional to mass of the body
The property of inertia is possessed by all bodies in nature
Intertial frame of reference , a = 0 (non accelrated) .
Newton’ ist law is about non accelrated frame of reference
Earth is considered to be an inertial frame of reference
Non inertial frame of reference, a > 0 or a < 0 (accelrated)
Newton’ 2nd law is also called law of accelration
F = ma (a inversly proportional to m) (a directly proportional to F) Example: A body of mass 2kg is moving with 2m/s² accelration. 2kg mass is added to the body. Now its accelration will be?? m1/m2 = a2/a1 ; 2/2+2 = a2/2 ; a2 = 1m/s²
A body is moving with constant acclration, the force acting on it will be constant
A body is moving with zero accelration, force acting on it will be zero
A body is moving with variable accelration, force acting on it will be variable
Newton’ 3rd law says every action has an equal and opposite reaction
Action and reaction forces never act on same body
The reaction force does not cancel the action force because they act on different bodies What is the effect on speed of a fighter plane chasing another when it opens fire? Speed of chasing plane ist decreases then increases according to newton 3rd law while speed of pursued plane increases when it returns the fire
Seat belts are useful in driving because these belts change the forces of motion prevent the passengers from moving forward when a moving car stops quickly
A motorcycle’s safety helmet is padded so as to extend time of injury of any collision to prevent serious injury
Momentum determines how easy or difficult it is to stop a moving body
The idea of linear momentum was introduced by Newton
P = mv Example: If % increase in velocity or mass of body is 70%, then % increase in momentum will also be 70% because there is direct relation without any square or underroot of v, m or P
SI unit is Ns
Momentum is not a dot or cross product of mass and velocity. It is just a product of mass and velocity
The rate of change of momentum(P/t) is equal to applied force Example: if velocity of a moving body of mass 2kg changes from 5m/s to 10m/s in 2sec, find force? Momentum =mvf-mvi = 20-10=10Ns ; F = P/t = 10/2 = 5N
Change of momentum takes place in direction of applied force
Second law in terms of linear momentum, “” force applied on body is equal to rate of change of momentum”””
K.E = mv²/2 = Pv/2
P = 2×K.E/v
K.E and momentum are equal if v=2m/s K.E = Pv/2 = P×2/2 = P
Product of force and time is called impulse
I = Ft = mv/t × t = mv
SI unit of impulse is Ns
Elastic collosion : total energy, k.e and linear momentum are conserved
Inelastic collision : total energy and linear momentum are conserved but k.e is not conserved
If two balls at same temperature collide, then momentum is conserved
When two objects of equal masses collide they inter change their velocities
When a ball of mass m moving with velocity v collides with exactly similar ball of mass m which is at rest, then after collision they inter change their velocities
If m2 >>> m1 and v2=0 then after collision m1 bounce back with same velocity while m2 remains stationary
If m1>>>m2 and v2=0 then after collision, there is no change in velocity of m1 while m2 begins to move with approximately twice of the velocity of m1
F = P/t = mv/t Example: Suppose Gases burn in rocket at 1000kg/s. Rocket eject gases at the rate of 2000km/s. Find force? F = v×m/t = 2×106 × 106 = 2×10¹²N
Suppose water flows out from pipe at 3kg/s and its velocity changes from 5m/s to zero on striking ball. Force = v × m/t = 5×3 = 15N
Propulsion of jet aircrafts, rocket and missile type weapons depends on application of momentum principles and law of action and reaction
Rocket equation is a = mv/M Here M is mass of rocket. When fuel is burned and ejected, mass of rocket decreases and hence accelration increases
A hose pipe ejects water at speed of 0.3m/s through a hole of area 50cm-2. If water strikes a wall normally, calculate force on wall, assuming velocity of water after striking wall is zero? F = Area × density × velocity² = 5×10-³×10³×0.3² = 0.45N
Projectile motion is 2 dimensional motion under constant accelration due to gravity
a = 0 in horizontal direction
F = 0 in horziontal direction
v remains unchanged in horizontal direction
S = Vx × t is horizontal distance
a = g in vertical direction
Y = vit + 1/2 gt² = (0)t + 1/2 gt² = 1/2 gt²
This vertical motion is same as for a freely falling body (vi=0)
Velocity along horizontal direction is Vfx = Vix = viCos theta
Velocity along vertical direction is Viy = viSin theta
Vfy = Viy- at = viSin theta – at
Tan theta = Vfy/Vfx = viSin theta – at/vix
At highest point of trajectory, velocity is minimum not zero
Height of projectile = vi² Sin² theta/2g
Time of flight = 2viSin theta/g
Time to reach maxi. height = viSin theta/g
Range of projectile = vi² Sin2 theta/g
For maximum range, theta = 45°
For maximum range, R = 4H
R = 4H when theta = 45°
For maximum height, R = 2H
R = 2H when theta = 63°
For Range = Height, theta = 76°
For determing ratio of range to height for any given angle, R/H = 4 [cos theta/sin theta]
If sum of two angles is 90°, these angles are said to be complementory angles
For complementory angles, ranges are equal For example at 60° and 30°, ranges are equal At 50° and 40°, ranges are equal
Horizontal range and maximum range of projectile are related as R = Rmax Sin2theta
Un-powered and un-guided missile is called ballistic missile
For short ranges and flat earth, trajectory is parabolic
For long ranges and curved(spherical) earth, trajectory is elliptical
The graph of displacement of a vertically thrown ball varrying with time is parabolic
If body is moving with uniform velocity then there will be no impulse
If body is moving with variable motion then there will be impulse
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