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Motion in a Straight Line

In physics, we are interested in understanding the motion of objects. Motion represents an object's continual change in position. Different types of motion exist: translational (movement from one point to another in a straight line), rotational (motion around an axis, e.g, an object spinning on its axis) and vibrational (repetitive and back and forth movement, e.g, a mass suspended by a spring). The motion in a straight line path is called kinematics. The simplest accelerated motion is straight line motion with constant acceleration. In this case the velocity changes at the same rate throughout the motion. This is a very special situation, yet one that occurs often in nature. Straight line motion with nearly constant acceleration also occurs in technology, such as a jet fighter being catapulted from the deck of an aircraft carrier.

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Distance & Displacement

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In everyday life, the terms distance and displacement are used as if they are the same but in physics, these two terms have different meanings. Distance is the length of actual path traversed. Distance is a scalar quantity, since it requires the statement of length only without specification of direction. For example, if a person moves 100 m from A to B and via path ACB and then returns from B to A, via path BDA covering 150 m; then we say distance traversed is 250m.  In length measurements, this is known as distance,which is simply the actual path length traveled. In length measurements, the straight line distance between the two points with a direction is called displacement. This is the shortest route between the initial and final points, with the direction of measurement specified.

Distance is a scalar quantity-magnitude (with units) only. Dispalcement is a vector quantity -magnitude (with units) and direction.

Distance is the actual length of path irrespective of direction. It is a scalar quantity.

Displacement is the shortest distance between initial and final positions and is expressed with direction. It is a vector quantity. The magnitude of displacement is always less than or equal to the distance.

The displacement of a body may be positive, negative or zero, depending upon the initial and final position of the moving body and its direction, while the magnitude of displacement is always positive. If the final position of body lies on the right of its initial position then the displacement of the body is taken positive, but if the final position is on the left of its initial position then the displacement is negative.

Instantaneous Velocity

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When the velocity of a particle is variable, we are generally interested in instantaneous velocity. The instantaneous velocity of a particle is the velocity of the particle at any instant of time or at any point of its path. Instantaneous velocity is the limiting value of average velocity  $\frac{\Delta \vec{x}}{\Delta t}$ as $\Delta t$ approaches zero. Its denoted by $\vec{v}$.

Instantaneous velocity,  $\vec{v}$=$\lim_{\Delta t\rightarrow 0}\frac{\Delta \vec{x}}{\Delta t}$

The magnitude of the instantaneous velocity is knows as instantaneous speed. It is given by

v = $\frac{dx}{dt}$

Uniform Motion with Examples

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When a body travels in a straight line and its velocity changes by equal amounts in equal intervals of time, it is said to have uniform acceleration. In other words ,a body is said to have uniform acceleration if the rate of change of its velocity remains constant. The motion of body with uniform acceleration is called uniformly accelerated motion. A body is said to have uniform motion if it travels equal distances in equal intervals of time, no matter how small these time intervals may be. 

Examples:
  • The movement of hands of watches
  • The movement of the earth about its axis
  • The movement of the earth around the sun
  • A gas molecule is in uniform motion between collision 
  • A body falling freely under gravity has uniform acceleration
  • A ball moving down an inclined plane has uniform acceleration

Non-uniform Motion with Examples

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When the velocity of a body changes by unequal amounts in equal intervals of time, it is said to have non-uniform acceleration. In other words, a body is said to have a non-uniform acceleration if the rate of change of its velocity is different at different points of time during its motion.  A body is said to have non uniform motion if it travels unequal distances in equal intervals of time however small these time intervals may be. Most of the motions, which we observe around us, are non uniform motions. 

Examples:
  • A train leaving a railway station covers larger and larger distances in equal intervals of time, conversely when it approaches a station, it covers smaller and smaller distances in equal time intervals.
  • A free falling stone under the action of gravity
  • When brakes are applied to a speeding car
  • When an oscillating simple pendulum is left for some time, the amplitude of its oscillation becomes smaller and smaller and finally the oscillation stops.
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