Motion In a Straight Line

Mechanics

• It is the branch of Physics, which deals with the study of motion of physical bodies.
• Mechanics can be broadly classified into following branches-

1. Statics 

It is the branch of mechanics, which deals with the study of physical bodies at rest

2. Kinematics

It is the branch of mechanics, which deals with study of motion of physical bodies without taking into account the factors, which cause motion.

3. Dynamics

It is the branch of mechanics, which deals with the study of motion of physical bodies taking into account the factors which cause motion.

Rest and Motion

1. Rest 

An object is said to be at rest if it does not changes its position with respect to the surrounding.

A Chair is said to be at rest with respect to the room.

2. Motion

An object is said to be in motion if it changes its position with respect to the surrounding.

When we ride or walk our body is in motion with respect to the surface or ground.

3. Rest and Motion are relative

Rest and motion depends upon the observer. The object in one situation may be at rest whereas the same object in another may be in motion.

Suppose you are traveling on a train that is moving at a high speed, and let's say there is another person observing you from outside the train. In this condition, you are in motion for that observer.

While studying the notes

1. We'll treat object as Point mass object.

2. An object can be considered as a point mass object if during the course of motion it covers distance greater than its own size.

3. We shall confine ourselves to the study of rectilinear motion.

4. Rectilinear motion is the study of motion of object along a Straight line.

4. Position, Distance and Displacement

1. Position

Position of object is always defined with respect to some reference point which we generally refer to as origin.

To define the change in position we have two physical quantities.

2. Distance 

• It is the actual path travelled by the body during the course of motion.
• The distance travelled by an object during the course of motion is never Negative or Zero and is always positive.
• The distance travelled is either equal or greater than displacement and is never less than magnitude of displacement ( Distance >= | Displacement | ) .
• It depends upon the path travelled.
• It is a Scalar Quantity.
• S.I Unit is 'm'.
• Dimensional unit is [ M⁰L¹T⁰ ] 

3. Displacement

• It is the difference between the final and initial position of the object during the course of motion.
• The displacement of an object may be positive, negative or, Zero during the course of motion.
• The magnitude of displacement is less than or equal to the distance travelled during the course of motion.
• The magnitude of displacement is independent of the path taken by an object during the course of motion.
• It is a Vector Quantity.
• S.I unit is 'm'.
• Dimensional unit is [ M⁰L¹T⁰ ] 

5. Speed and Velocity

1. Speed

• It is defined as the total path length travelled divided by the total time interval during which the motion has taken place.
• It is the Scalar Quantity.
• It is always positive during the course of the motion.
• It is greater than or equal to the magnitude of Velocity.
• The SI unit of speed is the meter per second (m/s). This unit represents the distance traveled in meters divided by the time taken in seconds.
• The dimensional formula for speed is [LT⁻¹].

2. Velocity

• It is defined as the change is position or displacement divided by the time interval, in which displacement occurs of.
• It is the Vector Quantity.
• It may be positive, negative or zero during the course of the motion.
• It is less than or equal to the speed.
• The SI unit of velocity is also the meter per second (m/s). Just like speed, it represents the rate of change of distance with respect to time.
• The dimensional formula for velocity is also [LT⁻¹].

If the motion of an object is along a straight line and in the same direction, the magnitude of displacement is equal to the total path length. In that case, the magnitude of average velocity is equal to the average speed. This is not always the case. The average velocity tells us how fast an object has been moving over a given interval but does not tell us how fast it moves at different intstants of time during that interval.

6. Scalar and Vector Quantities

1. Scalar Quantities

The physical quantities which have only magnitude but not direction, are called scalar quantities.

Examples : Mass, Length, Time, Distance, Speed, Work, Temperature.

2. Vector Quantities

The physical quantities which have magnitude as well as direction, are called vector quantities.

Example : Displacement, Velocity, Acceleration, Force, Momentum, Torque.

7. Average Velocity and Average Speed

1. Average Velocity

• It is defined as the change in position or displacement divided by the time intervals, in which displacement occurs.
• Its S.I unit is m/s, although Km/hr is used in many everyday applications.
• Dimensional unit is [LT⁻¹].

2. Average Speed

• It is defined as the total length travelled divided by the total time during which the motion has taken place.
• Its S.I unit is m/s.
• Dimensional unit is [LT⁻¹].

8. Instantaneous Velocity and Instantaneous Speed

1. Instantaneous Velocity

• It is velocity at an instant. The velocity at an instant is defined as the limit of the average velocity as the time interval Δt becomes infinitely small.
• Instantaneous velocity  = Lt(Δx / Δt) = dx / dt .
• The quantity on the right hand side of equation is the differential coefficient of x with respect to t and is denoted by dx / dt .
• It is the rate of change of position with respect to time at that instant.
• S.I unit is m/s.
• Dimensional unit is [LT⁻¹].

2. Instantaneous Speed

• Instantaneous speed or speed is the magnitude of velocity.
• S.I unit is m/s.
• Dimensional unit is [LT⁻¹].

9. Acceleration

1. Average Acceleration

• The average acceleration over a time interval is defined as the change of velocity divided by the time interval : a_avg = Δv / Δt .

Where: 

• a_avg represents the average acceleration.
• Δv represents the change in velocity. It is calculated as Δv = (v_f - v_i)
• Δt  represents the change in time. It is calculated as Δt = (t_f - t_i)

• It is the average change of velocity per unit time.
• S.I unit is m/s².
• Dimensional unit is [LT⁻²].

2. Instantaneous acceleration

• Instantaneous acceleration is defined in the same way as the instantaneous velocity : 
• a_{instantaneous} = lim_Δt→0 dv∕dt
• S.I unit is m/s^2.
• Dimensional unit is [LT⁻²].

• When the acceleration is uniform, obviously, instantaneous acceleration equals the average acceleration over that period.

• Since velocity is a quantity having both magnitude and direction, a change in velocity may involve either or both of these factor.

• Acceleration, therefore, may result from a change in the speed ( Magnitude ), a change in direction or changes in both.

• Like velocity, acceleration can also be positive, negative or zero.

10. Derivation of Equation of Motion by Integral method.

Derivation of \(v = u + at\):

\[v = u + at\]

\[a = \frac{dv}{dt}\]

\[ \int_{u}^{v} dv = \int_{0}^{t} a \, dt \]

\[ \left[ v \right]_{u}^{v} = \left[ at \right]_{0}^{t} \]

\[ v - u = at - 0 \]

\[ v - u = at \]

\[ v = u + at \]

Derivation of \(s = ut + \frac{1}{2}at^2\):

\[v = \frac{ds}{dt}\]

\[ \int_{0}^{s} ds = \int_{0}^{t} (u + at) \, dt \]

\[ \left[ s \right]_{0}^{s} = \left[ ut + \frac{1}{2}at^2 \right]_{0}^{t} \]

\[ s - 0 = ut + \frac{1}{2}at^2 - 0 \]

\[ s = ut + \frac{1}{2}at^2 \]

Derivation of \(v^2 = u^2 + 2as\):

\[a = \frac{dv}{dt}\]

\[a = \frac{dv}{dt} \cdot \frac{ds}{ds}\]

\[a = \frac{dv}{ds} \cdot \frac{ds}{dt}\]

\[a = v \cdot \frac{dv}{ds}\]

\[ \int_{0}^{s} a \, ds = \int_{u}^{v} v \cdot \frac{dv}{ds} \, ds \]

\[ as = \left[\frac{1}{2}v^2\right]_{u}^{v} \]

\[ as = \frac{1}{2}v^2 - \frac{1}{2}u^2 \]

\[ as + \frac{1}{2}u^2 = \frac{1}{2}v^2 \]

\[ v^2 = u^2 + 2as \]