TOPIC-1
Kinematics :
- Kinematics is the study of motion of physical bodies without going into the cause of the motion.
- Kinematics deals with physical quantities like distance, displacement, speed, velocity, acceleration etc.
Motion and Rest :
- Motion is a combined property of the object under study and observer.
- If the position of the object under study changes with time, as seen by the observer, the object is said to be in motion from the frame of reference of the observer.
- If position of the object does not change with time, as seen by observer, object is said to be at rest from the frame of reference of the observer.
- Rest and motion of an object under study depend on the frame of reference of the observer. For eg. A book kept on a table may be at rest for all students sitting in the class. But the same book will be in motion, as seen by an observer on a moving bus. Thus absolute rest and absolute motion are meaningless.
- In most cases, if attributes of motion of an object are given without specifying the frame of the observer, it is to be assumed that the object under consideration is being observed by an observer who is at rest with respect to the earth.
Position :
- For a particle moving along a straight line, position of the particle can be specified with only one coordinate. A coordinate system is chosen by choosing some reference point as the origin. The origin is assigned the number zero. Most situations can be analysed by setting up an appropriate coordinate system. In order to do so, following are the essential requirements:
- Choice of origin
- Choice of coordinate axis
- Choice of positive direction of axis. All these parameters constitute a reference frame. In any physics problem, the reference frame must be specified.
- In the figure below, point O is the chosen origin, X - axis is the chosen coordinate axis and rightward direction is chosen as the positive direction.
- Similarly, if the motion of a particle is 2 - dimensional or 3 - dimensional, the coordinate axes will comprise of x, y and z axes and position will include x, y, and z coordinates.
Displacement and distance :
- Displacement is a vector quantity. It is the change in position vector. Distance is the total length of the actual path covered. Distance is a scalar quantity.
- Suppose a particle travels from point A to point B as shown in the fig below along a zig -zag path in a finite time interval.
Coordinates of A are( \({x_1},{y_1}\) )and that of B are ( \({x_2},{y_2}\) ) . Position vector of A is\({{\vec r}_A} = {x_1}\hat i + {y_1}\hat j\) ,
position of vector of B is \({{\vec r}_B} = {x_2}\hat i + {y_2}\hat j\). Distance will be equal to the total length of the actual path covered by the particle. Displacement will be \(\vec S = {{\vec r}_B} - {{\vec r}_A} = \left( {{x_2} - {x_1}} \right)\hat i + \left( {{y_2} - {y_1}} \right)\hat j\).
- The distance covered will always be greater than or equal to the magnitude of the displacement.
- Displacement and distance are equal in magnitude in case the particle is travelling along a straight line without change in direction.
- SI unit of distance and displacement is meters.
- In simple language, displacement can be said to be the shortest line joining the initial and final positions of a body in motion, irrespective of path followed and it is directed from initial position to final position.
- Change in position vector is displacement and change in displacement vector is also displacement.
Average speed :
- \(AvgSpeed = \frac{{TotalDis\tan ceTravelled}}{{TotalTime}}\) ,We define average speed of a particle as the ratio of the total distance travelled to the total time taken.
- SI units of speed is m / s
Instantaneous speed :
- Speed of a particle at a particular instant is called instantaneous speed.
- The speedometer of a vehicle indicates the instantaneous speed. The speedometer reading on a crowded city road continuously changes, indicating instantaneous speed is continuously changing.
Velocity :
- Velocity is defined as rate of change of displacement with time.Velocity is a vector quantity. SI unit of velocity is m/s.
- \(AvgVelocity = \frac{{TotalDisplacement}}{{TotalTime}}\) , Average velocity is defined as the ratio of the Total time displacement covered to the total time taken.
- Just as distance is always greater than or equal to the magnitude of displacement, average speed is greater than or equal to the magnitude of average velocity. Average speed and the magnitude of average velocity are equal when particle is travelling in a straight line without change in direction.
Instantaneous velocity :
- Suppose a particle moves from position x at time t to position \(x + \Delta x\) at time \(t + \Delta t\) . Then, the average velocity of the particle over time interval \(\Delta t\) is \(\frac{{\Delta x}}{{\Delta t}}\).
- Making \(\Delta t\)infinitely small,\(\frac{{\Delta x}}{{\Delta t}}\) gives the velocity of the particle at instant t and can be written as\(v = \mathop {\lim }\limits_{\Delta t \to 0} \frac{{\Delta x}}{{\Delta t}} = \frac{{dx}}{{dt}}\),where v is the instantaneous velocity of the particle at time instant t.
- The magnitudes of instantaneous velocity and instantaneous speed are always equal.
Uniform motion :
- Motion of a body in a straight line with uniform velocity is called uniform motion.\(\frac{{ds}}{{dt}} = v\), but v is constant in uniform motion.\(\therefore \int {ds = } \int {vdt = v\int {dt = vt} } \)
- Uniform motion can also be said to be motion in which equal displacements are covered in equal intervals of time, however small the time intervals may be.
