Mechanics Kinematics Introduction

Kinematics is a branch of Physics which deals with motion of a body with out considering the forces acting on it.

The study of motion of objects without any reference to the cause of motion is called kinematics.

Distance and Displacement

1. The actual path length traversed by a body is called the distance travelled.
2. The shortest distance between the initial and final positions of a body is called displacement.
3. Displacement of a body may be zero, or positive or negative but distance travelled is always positive.
4. If a particle covers a distance d and displacement s between two points then s<=d in magnitude.
5. The speed of a body is the rate at which it describes its path.
6. The rate of change of position is called velocity.
7. Average speed = total distance / total time
8. Average velocity = net displacement / total time

Mechanics linear motion :

Motion of a body along a line is called linear motion. Here average velocity is total displacement divided by the total time.

If a body travels with a velocity v1 for the first half of the journey time and with a velocity v2 for the second half of the journey time, then the average velocity v₁+V<sub>2/2

If a body covers first half of its journey with uniform velocity v1 and the second half of the journey with uniform velocity v2, then the average velocity V = 2</sub>v₁V2/v₁+V2.

The rate of change of velocity is called acceleration.

Please click the below image to see further important points about linear motion and its equations under different circumstances.

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Projectile Motion

It is nothing but two dimensional motion.It could be either from ground or from a certain height.If a body is changing its position along X and Y axis with respect to time then it is called projectile motion.

An object that is in flight after being thrown or projected is called a projectile. Such a projectile might be a football, a cricket ball, a baseball or any other object. The motion of a projectile may be thought of as the result of two separate, simultaneously occurring components of motions. One component is along a horizontal direction without any acceleration and the other along the vertical direction with constant acceleration due to the force of gravity.

we shall assume that the air resistance has negligible effect on the motion of the projectile for a simple projectile equation.

Let us assume that the projectile is launched with velocity vo that makes an angle θo with the x-axis as shown in Fig. below.After the object has been projected, the acceleration acting on it is that due to gravity which is directed vertically downward:

a = −g j which means that
ax = 0, ay = – g
The components of initial velocity vo are :

v along x axis is = vo cos θo
v along y axis is = vo sin θo

If we take the initial position to be the origin of the reference frame as shown in Fig. we have :

One of the components of velocity, i.e. x-component remains constant throughout the motion and only the y- component changes, like an object in free fall in vertical direction. This is shown graphically at few instants in Fig. below Note that at the point of maximum height, vy= 0 and therefore, tan θ = y component of velocity /x component of velocity = 0.

The equation for the path of a projectile can be derived as shown below.

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Kinematics Linear Motion

Previously we have discussed regarding speed,velocity, acceleration of one dimensional motion.Here we are going to extend that more in detail and going through the above posts will definitely help in understanding the present concepts.

Basic definitions of Kinematics :

1. The study of motion of objects without any reference to the cause of motion is called kinematics.
2. The actual path traversed by a body is called the distance traveled.
3. The shortest distance between the initial and final positions of a body is called displacement.
4. Displacement of a body may be zero, or positive or negative but distance traveled is always positive.
5. The speed of a body is the rate at which it describes its path.
6. The rate of change of displacement is called velocity.
7. Average speed = total distance / total time
8. Average velocity = net displacement / total time

Linear Motion :

It is nothing but one dimensional motion where body always moves along a line.We can derive the following formula in the case of linear motion.

1 . Average velocity V = Total displacement / Total time

Avg.velocity = s₁+s₂+s₃/t₁+t₂+t₃ where t₁ is time in the first case where as s₁ is the displacement in the first case and so on.

2 . If a body travels with a velocity v1 for the first half of the journey time and with a velocity v2 for the second half of the journey time, then the average velocity is equal to v₁+v₂/2.

3 . If a body covers first half of its journey with uniform velocity v1 and the second half of the journey with uniform velocity v2, then the average velocity is equal to 2v₁v₂/v₁+v₂ .

4 . If a body travels first one third of the distance with a speed v₁, and second one third of the distance with a speed v₂ and the last one third of the distance with a speed v₃ then the average velocity is 3v₁v₂v₃/v₁v₂+v₂v₃+v₃v₁.

5. The rate of change of velocity is called acceleration.

Equations of motion for a body moving with uniform acceleration

The following equations represent the motion of a body under constant acceleration.

If a body starts from rest and having uniform acceleration then the above equations can be modified as shown below.

Please click on the screen for a better view.


Notes :

1 . If the velocity of a body becomes 1/n of original velocity after a displacement x then it will come to rest after covering a further displacement of X/n² - 1 .

2 . A body is describing uniform circular motion with a speed ‘v’. When it describes an angle ‘q’ at the center then the change in velocity is dv = 2vsin (q/2)

3 . If the displacement of a body is proportional to the square of time, then its initial velocity is zero.

4 . Starting from rest a body travels with an acceleration ‘a’ for some time and then with deceleration ‘b’ and finally comes to rest. If the total time of journey is ‘t’, then the maximum velocity and displacement and average velocity are respectively then

i) Maximum velocity = ab t/a + b
ii) Displacement s = abt²/2(a+b).
iii)Average Velocity = maximum velocity / 2.

5 . If a particle starts from rest and moves with uniform acceleration ‘a’ such that it travels distances X and Y in the m and n particular seconds then

s_n/s = X-Y/m-n where n is the particular second of journey.

6 . A particle starts from rest and moves along a straight line with uniform acceleration. If s is the distance traveled by it n seconds and S is the distance travel led in the particular n th second then s_n/s = 2n-1 /n²

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Kinematics Accelaration

The previous post about kinematics deals with SPEED AND VELOCITY and it can be browsed here.

Acceleration as the rate of change of velocity with time.

The average acceleration `a over a time interval is defined as the change of velocity divided by the time interval :

`a = D v /Dt .

where v2 and v1 are the instantaneous velocities or simply velocities at time t2 and t1. It is the average change of velocity per unit time. The SI unit of acceleration is m s–2 .

On a plot of velocity versus time, the average acceleration is the slope of the straight line connecting the points corresponding to (v2, t2) and (v1, t1). The average acceleration for velocity-time graph shown in figure for different time intervals 0 s - 10 s, 10 s – 18 s, and 18 s and – 20 s are :

0 s - 10 s `a = 24 - 0 /10 - 0 = 2.4 m s–2 .

10 s - 18 s `a = 24 - 24 / 18 - 10 = 0 m s–2 .

18 s - 20 s `a = 0 - 24 / 20 - 18 = -12 m s–2.

Instantaneous acceleration is defined as rate of change of velocity at any particular instant.It is shown as a = dv/dt .The acceleration at an instant is the slope of the tangent to the v–t curve at that instant.

Since velocity is a quantity having both magnitude and direction, a change in velocity may involve either or both of these factors. Acceleration, therefore, may result from a change in speed (magnitude), a change in direction or changes in both. Like velocity, acceleration can also be positive, negative or zero.

Position-time graphs for motion with positive, negative and zero acceleration are shown in figures (a), (b) and (c), respectively.The graph curves upward for positive acceleration; downward for negative acceleration and it is a straight line for zero acceleration.

The area under the curve represents the displacement over a given time interval.of velocity-time graph for any moving object .

Let us check for a simple case as follows.

Let an object moving with constant velocity u. Its velocity-time graph is as shown in figure.

The v-t curve is a straight line parallel to the time axis and the area under it between t = 0 and t = T is the area of the rectangle of height u and base T. Therefore, area = u × T = uT which is the displacement in this time interval.

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Kinematics Velocity and Speed

The previous post in kinematics deals with one dimensional motion concept and you can browse it here.
AVERAGE VELOCITY AND AVERAGE SPEED :

Average velocity is defined as the change in position or displacement (Δx) divided by the time intervals (Δt),

v = X_{2 -} X1/t_{2 -} t1

where x2 and x1 are the positions of the object at time t2and t1, respectively. Here bar over the symbol for velocity is a standard notation used to indicate an average quantity. The SI unit for velocity is m/s or , although km/h is used in many everyday applications.

The average velocity can be positive or negative depending upon the sign of the displacement. It is zero if the displacement is zero. The following fugures shows the x-t graphs for an object, moving with positive velocity (Fig. a), moving with negative velocity (Fig. b) and at rest (Fig. c).

Average speed is defined as the total path length travelled divided by the total time interval during which the motion has taken place :

Average speed = Total path length/Total time interval

Average speed has obviously the same unit(m/s) as that of velocity. But it does not tell us in what direction an object is moving. Thus, it is always positive (in contrast to the average velocity which can be positive or negative). 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.

INSTANTANEOUS VELOCITY AND SPEED :The velocity at an instant is defined as the limit of the average velocity as the time interval Δt becomes infinitesimally small.

v = dx/dt
For uniformmotion, velocity is the same as the average velocity at all instants.

Instantaneous speed or simply speed is the magnitude of velocity.

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One Diemensional Motion concepts part one

Motion is change in position of an object with time.Study of motion of objects along a straight line is known as rectilinear motion.

FRAME OF REFERENCE:

we need to use a reference point and a set of axes. It is convenient to choose a rectangular coordinate system consisting of three mutually perpendicular axes, labeled X-, Y-, and Z- axes.

The point of intersection of these three axes is called origin (O) and serves as the reference point. The coordinates (x, y. z) of an object describe the position of the object with respect to this coordinate system.

To measure time, we position a clock in this system. This coordinate system along with a clock constitutes a frame of reference.

Description of an event depends on the frame of reference chosen for the description. For example, when you say that a car is moving on a road, you are describing the car with respect to a frame of reference attached to you or to the ground. But with respect to a frame of reference attached with a person sitting in the car, the car is at rest.

To describe motion along a straight line, we can choose an axis, say X-axis, so that it coincides with the path of the object.

Displacement has both magnitude and direction. Such quantities are represented by vectors.

If one or more coordinates of an object change with time, we say that the object is in motion. Otherwise, the object is said to be at rest with respect to this frame of reference.

The choice of a set of axes in a frame of reference depends upon the situation. For example, for describing motion in one dimension, we need only one axis. To describe motion in two/three dimensions, we need a set of two/ three axes.

The magnitude of displacement may or may not be equal to the path length traversed by an object.

The magnitude of the displacement for a course of motion may be zero but the corresponding path length is not zero. For example, if the car starts from O, goes to P and then returns to O, the final position coincides with the initial position and the displacement is zero. However, the path length of this journey is OP + PO.

Motion of an object can be represented by a position-time graph as you have already learnt about it. Such a graph is a powerful tool to represent and analyse different aspects of motion of an object. For motion along a straight line, say X-axis, only x-coordinate varies with time and we have an x-t graph. Let us first consider the simple case in which an object is stationary, e.g. a car standing still at x = 40 m. The position-time graph is a straight line parallel to the time axis, as shown in Fig.

If an object moving along the straight line covers equal distances in equal intervals of time, it is said to be in uniform motion along a straight line.The following figure shows the position-time graph of such a motion.


Now, let us consider the motion of a car that starts from rest at time t = 0 s from the origin O and picks up speed till t = 10 s and thereafter moves with uniform speed till t = 18 s. Then the brakes are applied and the car stops at t = 20 s and x = 296 m. The position-time graph for this case is shown in Figure below.

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