Acceleration
Category : JEE Main & Advanced
The time rate of change of velocity of an object is called acceleration of the object.
(1) It is a vector quantity. It?s direction is same as that of change in velocity (Not of the velocity)
Possible ways of velocity change
| When only direction of velocity changes | When only magnitude of velocity changes | When both magnitude and direction of velocity changes |
| Acceleration perpendicular to velocity | Acceleration parallel or anti-parallel to velocity | Acceleration has two components one is perpendicular to velocity and another parallel or anti-parallel to velocity |
| Ex.. Uniform circular motion | Ex.. Motion under gravity | Ex.. Projectile motion |
(2) Dimension : \[[{{M}^{0}}{{L}^{1}}{{T}^{-2}}]\]
(3) Unit : metre/second2 (S.I.); cm/second2 (C.G.S.)
(4) Types of acceleration :
(i) Uniform acceleration : A body is said to have uniform acceleration if magnitude and direction of the acceleration remains constant during particle motion.
(ii) Non-uniform acceleration : A body is said to have non-uniform acceleration, if either magnitude or direction or both of them change during motion.
(iii) Average acceleration : \[{{\vec{a}}_{a\upsilon }}=\frac{\Delta \vec{v}}{\Delta t}=\frac{{{{\vec{v}}}_{2}}-{{{\vec{v}}}_{1}}}{\Delta t}\]
The direction of average acceleration vector is the direction of the change in velocity vector as \[\vec{a}=\frac{\Delta \vec{v}}{\Delta t}\]
(iv) Instantaneous acceleration = \[\vec{a}=\underset{\Delta t\to 0}{\mathop{\lim }}\,\frac{\Delta \vec{v}}{\Delta t}=\frac{d\vec{v}}{dt}\]
(v) For a moving body there is no relation between the direction of instantaneous velocity and direction of acceleration.
Ex.. (a) In uniform circular motion \[\theta ={{90}^{o}}\] always
(b) In a projectile motion \[\theta \] is variable for every point of trajectory.
(vi) If a force \[\overrightarrow{F}\] acts on a particle of mass m, by Newton's 2nd law, acceleration \[\vec{a}=\frac{{\vec{F}}}{m}\]
(vii) By definition \[\vec{a}=\frac{d\vec{v}}{dt}=\frac{{{d}^{2}}\vec{x}}{d{{t}^{2}}}\]\[\left[ \text{As}\,\,\vec{v}=\frac{d\vec{x}}{dt} \right]\]
i.e., if x is given as a function of time, second time derivative of displacement gives acceleration
(viii) If velocity is given as a function of position, then by chain rule \[a=\frac{dv}{dt}=\frac{dv}{dx}\times \frac{dx}{dt}=v.\frac{d\upsilon }{dx}\,\left[ \text{as}\,\,v=\frac{dx}{dt} \right]\]
(ix) Acceleration can be positive, zero or negative. Positive acceleration means velocity increasing with time, zero acceleration means velocity is uniform constant while negative acceleration (retardation) means velocity is decreasing with time.
(x) For motion of a body under gravity, acceleration will be equal to "g", where g is the acceleration due to gravity. Its value is \[9.8\,\,\text{m/}{{\text{s}}^{\text{2}}}\] or \[980\,\,\text{cm/}{{\text{s}}^{\text{2}}}\] or \[32\,\,\text{feet/}{{\text{s}}^{\text{2}}}\].
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