A particle is moving in a straight line such that its distance at any time $t$ is given by $S=\frac{t^{4}}{4}-2 t^{3}+4 t^{2}-7$. Find when its velocity is maximum and acceleration minimum.
Given : $s=\frac{t^{4}}{4}-2 t^{3}+4 t^{2}-7$
$\Rightarrow v=\frac{d s}{d t}=t^{3}-6 t^{2}+8 t$
$\Rightarrow a=\frac{d v}{d t}=3 t^{2}-12 t+8$
For maximum or minimum values of $v$, we must have
$\frac{d v}{d t}=0$
$\Rightarrow 3 t^{2}-12 t+8=0$
On solving the equation, we get
$t=2 \pm \frac{2}{\sqrt{3}}$
Now,
$\frac{d^{2} v}{d t^{2}}=6 t-12$
At $t=2-\frac{2}{\sqrt{3}}:$
$\frac{d^{2} v}{d t^{2}}=6\left(2-\frac{2}{\sqrt{3}}\right)-12$
$\Rightarrow \frac{-12}{\sqrt{3}}<0$
So, velocity is maximum at $t=\left(2-\frac{2}{\sqrt{3}}\right)$.
Again,
$\frac{d a}{d t}=6 t-12$
For maximum or minimum values of a, we must have
$\frac{d a}{d t}=0$
$\Rightarrow 6 t-12=0$
$\Rightarrow t=2$
Now,
$\frac{d^{2} a}{d t^{2}}=6>0$
So, acceleration is minimum at $t=2$.
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,