# Motion

**Definition**

*If an object continuously changes its position with respect to its surrounding, then it is said to be in state of motion*.

**Rectilinear Motion**

The motion along a straight line is called rectilinear motion.

## Velocity

Velocity may be defined as the change of displacement of a body with respect time.

Velocity = change of displacement / time

Velocity is a vector quantity and its unit in S.I system is meter per second (m/sec).

**Average Velocity**

Average velocity of a body is defined as the ratio of the displacement in a certain direction to the time taken for this displacement.

Suppose a body is moving along the path AC as shown in figure. At time t1, suppose the body is at P and its position w.r.t origin O is given by vector r2.

Thus, displacement of the body = r2 - r1 = Δr

Time taken for this displacement - t2 - t1 = Δt

Therefore, average velocity of the body is given by

Vav = Δr / Δt

**Instantaneous Velocity**

It is defined as the velocity of a body at a certain instant.

V(ins) = 1im Δr / Δt

Where Δt → 0 is read as "Δt tends to zero", which means that the time is very small.

**Velocity From Distance - Time Graph**

We can determine the velocity of a body by distance - time graph such that the time is taken on x-axis and distance on y-axis.

## Acceleration

Acceleration of a body may be defined as the time rate of change velocity. If the velocity of a body is changing then it is said to posses acceleration.

Acceleration = change of velocity / time

If the velocity of a body is increasing, then its acceleration will be positive and if the velocity of a body is decreasing, then its acceleration will be negative. Negative acceleration is also called *retardation*.

Acceleration is a vector quantity and its unit in S.I system is meter per second per second. (m/sec2 OR m.sec-2)

**Average Acceleration**

Average acceleration is defined as the ratio of the change in velocity of a body and the time interval during which the velocity has changed.

Suppose that at any time t1 a body is at A having velocity V1. At a later time t2, it is at point B having velocity V2. Thus,

Change in Velocity = V2 - V1 = Δ V

Time during which velocity has changed = t2 - t1 = Δ t

**Instantaneous Acceleration**

It is defined as the acceleration of a body at a certain instant

a(ins) = lim Δ V / Δ t

where Δt → 0 is read as "Δt tends to zero", which means that the time is very small.

**Acceleration from Velocity - Time Graph**

We can determine the acceleration of a body by velocity - time graph such that the time is taken on x-axis and velocity on y-axis.

**Equations of Uniformly Accelerated Rectilinear Motion**

There are three basic equations of motion. The equations give relations between

Vi = the initial velocity of the body moving along a straight line.

Vf = the final velocity of the body after a certain time.

t = the time taken for the change of velocity

a = uniform acceleration in the direction of initial velocity.

S = distance covered by the body.

Equations are

1. Vf = Vi + a t

2. S = V i t + 1/2 a t2

3. 2 a S = V f2 - V i 2

**Motion Under Gravity**

The force of attraction exerted by the earth on a body is called gravity or pull of earth. The acceleration due to gravity is produced in a freely falling body by the force of gravity. Equations for motion under gravity are

1. Vf = Vi + g t

2. S = V i t + 1/2 g t2

3. 2 g S = Vr2 - Vi2

where g = 9.8 m / s2 in S.I system and is called acceleration due to gravity.

## Law of Motion

Isaac Newton studied motion of bodies and formulated three famous laws of motion in his famous book "Mathematical Principles of Natural Philosophy" in 1687. These laws are called Newton's Laws of Motion.

## Newton's First Law of Motion

**Statement**

*A body in state of rest will remain at rest and a body in state of motion continues to move with uniform velocity unless acted upon by an unbalanced force*.

**Explanation**

This law consists of two parts. According to first part a body at rest will remain at rest will remain at rest unless some external unbalanced force acts on it. It is obvious from our daily life experience. We observe that a book lying on a table will remain there unless somebody moves it by applying certain force. According to the second part of this law a body in state of uniform motion continuous to do so unless it is acted upon by some unbalanced force.

This part of the law seems to be false from our daily life experience. We observe that when a ball is rolled in a floor, after covering certain distance, it stops. Newton gave reason for this stoppage that force of gravity friction of the floor and air resistance are responsible of this stoppage which are, of course, external forces. If these forces are not present, the bodies, one set into motion, will continue to move for ever.

**Qualitative Definition of Net Force**

The first law of motion gives the qualitative definition of the net force. (Force is an agent which changes or tends to change the state of rest or of uniform motion of a body).

**First Law as Law of Inertia**

Newton's first law of motion is also called the Law of inertia. *Inertia* is the property of matter by virtue of which is preserves its state of rest or of uniform motion. Inertia of a body directly related to its mass.

## Newton's Second Law of Motion

**Statement**

*If a certain unbalanced force acts upon a body, it produces acceleration in its own direction. The magnitude of acceleration is directly proportional to the magnitude of the force and inversely proportional to the mass of the body*.

**Mathematical Form**

According to this law

f ∞ a

F = m a → Equation of second law

Where 'F' is the unbalanced force acting on the body of mass 'm' and produces an acceleration 'a' in it.

From equation

1 N = 1 kg x 1 m/sec2

Hence one newton is that unbalanced force which produces an acceleration of 1 m/sec2 in a body of mass 1 kg.

**Vector Form**

Equation of Newton's second law can be written in vector form as

F = m a

Where F is the vector sum of all the forces acting on the body.

## Newton's Third Law of Motion

**Statement**

*To every action there is always an equal and opposite reaction*.

**Explanation**

For example, if a body A exerts force on body B (F(A) on B) in the opposite direction. This force is called *reaction*. Then according to third law of motion.

**Examples**

1. When a gun is fired, the bullet flies out in forward direction. As a reaction of this action, the gun reacts in backward direction.

2. A boatman, when he wants to put his boat in water pushes the bank with his oar, The reaction of the bank pushes the boat in forward direction.

3. While walking on the ground, as an action, we push the ground in the backward direction. As a reaction ground pushes us in the forward direction.

4. In flying a kite, the string is given a downward jerk and is then released. Thereupon the reaction of the air pushes the kite upward and makes it rise higher.

**Tension in a String**

Consider a body of weight W supported by a person with the help of a string. A force is experienced by the hand as well as by the body. This force is known as *Tension*. At B the hand experiences a downward force. So the direction of force at point B is downward. But at point A direction of the force is upward.

These forces at point A and B are tensions. Its magnitude in both cases is same but the direction is opposite. At point A,

Tension = T = W = mg

## Momentum of a Body

The momentum of a body is the quantity of motion in it. It depends on two things

1. The mass of the object moving (m),

2. The velocity with which it is moving (V).

Momentum is the product of mass and velocity. It is denoted by P.

P = m V

Momentum is a vector quantity an its direction is the same as that of the velocity.

**Unit of Momentum**

Momentum = mass x velocity

= kg x m/s

= kg x m/s x s/s

= kg x m/s2 x s

since kg. m/s2 is newton (N)

momentum = N-s

Hence the S.I unit of momentum is N-s.

**Unbalanced or Net Force is equal to the Rate of Change of Momentum**

*i.e., F = (mVf = mVi) / t*

**Proof**

Consider a body of mass 'm' moving with a velocity Vl. A net force F acts on it for a time 't'. Its velocity then becomes Vf.

Therefore

Initial momentum of the body = m Vi

Final momentum of the body = m Vf

Time interval = t

Unbalanced force = F

Therefore

Rate of change of momentum = (m Vf - m Vi) / t ....................... (1)

But

(Vf - Vi) / t = a

Therefore,

Rate of change of momentum = m a = F ..................... (2)

Substituting the value of rate of change of momentum from equation (2) in equation (1), we get

F = (m Vf - m Vi) / t ............................. Proved

## Law of Conservation of Momentum

**Isolated System**

When a number of bodies are such that they exert force upon one another and no external agency exerts a force on them, then they are said to constitute and isolated system.

**Statement of the Law**

*The total momentum of an isolated system of bodies remains constant*.

**OR**

*If there is no external force applied to a system, then the total momentum of that system remains constant*.

**Elastic Collision**

An elastic collision is that in which the momentum of the system as well as the kinetic energy of the system before and after collision, remains constant. Thus for an elastic collision.

If P momentum and K.E is kinetic energy.

P(before collision) = P(after collision)

K.E(before collision) = K.E(after collision)

**Inelastic Collision**

An inelastic collision is that in which the momentum of the system before and after the collision remains constant but the kinetic energy before and after the collision changes.

Thus for an inelastic collision

P(before collision) = P(after collision)

**Elastic Collision in one Dimension**

Consider two smooth non rotating spheres moving along the line joining their centres with velocities U1 and U2. U1 is greater than U2, therefore the spheres of mass m1 makes elastic collision with the sphere of mass m2. After collision, suppose their velocities become V1 and V2 but their direction of motion is along same line as before.

## Friction

When two bodies are in contact, one upon the other and a force is applied to the upper body to make it move over the surface of the lower body, an opposing force is set up in the plane of the contract which resists the motion. This force is the force of friction or simply friction.

The force of friction always acts parallel to the surface of contact and opposite to the direction of motion.

**Definition**

*When one body is at rest in contact with another, the friction is called Static Friction*.

*When one body is just on the point of sliding over the other, the friction is called Limiting Friction*.

When one body is actually sliding over the other, the friction is called Dynamic Friction*.*

**Coefficient of Friction (μ)**

The ratio of limiting friction 'F' to the normal reaction 'R' acting between two surfaces in contact is called the coefficient of friction (μ).

μ = F / R

**Or**

F = μ R

**Fluid Friction**

Stoke found that bodies moving through fluids (liquids and gases) experiences a retarding force fluid friction or viscous drag. If the moving bodies are spheres then fluid friction F is given by

F = 6 π η r v

Where η is the coefficient of viscosity,

Where r is the radius of the sphere,

Where v is velocity pf the sphere.

**Terminal Velocity**

When the fluid friction is equal to the downward force acting on the sphere, the sphere attains a uniform velocity. This velocity is called Terminal velocity.

## The Inclined Plane

A plane which makes certain angle θ with the horizontal is called an inclined plane.

Consider a block of mass 'm' placed on an inclined plane making certain angle θ with the horizontal. The forces acting on the block are

1. W, weight of the block acting vertically downward.

2. R, reaction of the plane acting perpendicular to the plane

3. f, force of friction which opposes the motion of the block which is moving downward.

Now we take x-axis along the plane and y-axis perpendicular to the plane. We resolve W into its rectangular components.

Therefore,

Component of W along x-axis = W sin θ

And

Component of W along y-axis = W cos θ

*1. If the Block is at Rest*

According to the first condition of equilibrium

Σ Fx = 0

Therefore,

f - W sin θ = 0

**Or**

f = W sin θ

Also,

Σ Fy = 0

Therefore,

R - W cos θ = 0

**Or**

R = W cos θ

*2. If the Block Slides Down the Inclined Plane with an Acceleration*

Therefore,

W sin θ > f

Net force = F = W sin θ - f

Since F = m a and W = m g

Therefore,

m a = m g sin θ - f

*3. When force of Friction is Negligible*

Then f ≈ 0

Therefore,

equation (3) => m a = m g sin ≈ - 0

=> m a = m g sin ≈

or a = g sin ≈ ............. (4)

**Particular Cases**

*Case A : If the Smooth Plane is Horizontal Then 0 = 0º*

Therefore,

Equation (4) => a = g sin 0º

=> a = g x 0

=> a = 0

*Case B : If the Smooth Plane is Vertical Then θ = 90º*

Therefore,

Equation (4) => a = g sin 90º

=> a = g x 1

=> a = g

This is the case of a freely falling body.

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