A Particle moves along a straight line such that its displacement at any time t is given by
s = (t^3 - 6t^2 + 3t +4) m
The velocity when the acceleration is zero.
Answers
Given :
▪ Displacement-time equation of a particle has been provided.
To Find :
▪ Velocity of particle when the acceleration is zero.
SoluTion :
Given,
Therefore,
Acceleration a is given by
Hence, at t = 2 s the velocity will be
Answer- The above question is from the chapter 'Kinematics'.
Some important terms and formulae:-
1. Velocity- It is the displacement per unit time.
S.I. Unit of Velocity- m/s
It is a vector quantity as it possesses magnitude and direction.
2. Acceleration- It is the rate of change of velocity.
S.I. Unit of Acceleration- m/s²
It is also a vector quantity.
Negative acceleration is called retardation.
3. Distance- It is the path length transversed by an object.
S.I. Unit of Distance- m
It is a scalar quantity.
4. Displacement- It is the shortest distance between the initial and final point.
S.I. Unit of Displacement- m
It is a vector quantity.
5. Equations for uniformly accelerated motion-
Let u = Initial velocity of a particle
v = Final velocity of a particle
t = Time taken
s = Distance travelled in the given time
a = Acceleration
1) v = u + at
2) s = at² + ut
3) v² - u² = 2as
6. Average Speed = Total distance ÷ Total time
7. Average Velocity = Total displacement ÷ Total time
8. Some basic formulae using calculus method:
1) Velocity =
2) Acceleration =
3) Acceleration (when x is given) =
Given question: A particle moves along a straight line such that its displacement at any time t is given by s = (t³ - 6t² + 3t + 4) m.
Fine the velocity when the acceleration is zero.
Answer: s = (t³ - 6t² + 3t + 4) m
We know that v =
On differentiating with respect to t, we get,
v = (3t² - 12t + 3) m/s
We also know that, a = .
Again on differentiating with respect to t, we get,
a = (6t - 12) m/s²
It is given that a = 0 m/s²
⇒ 0 = 6t - 12
⇒ 6t = 12
⇒ t = 2 seconds
So, substituting the value of t in v, we get,
v = [3(2)² - 12(2) + 3] m/s
v = [12 - 24 + 3] m/s
v = - 9 m/s