- 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 = 2×10*6 × 10*6 = 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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