Question

Position of a particle at any instant is given as \[3\sin (\pi t)\]. So, find its speed at t = 1 second its acceleration.

Answer 1

Hint: We have been given the position of a particle as a function of time. We have been asked to calculate the speed and acceleration of the particle. Now, we know that speed is the rate of change of position and acceleration is the rate of change of velocity. Therefore, we will be using this to calculate the speed and acceleration of the given particle.

Formula Used:
 \[v=\dfrac{dx}{dt}\]
Where,
x is the position of the particle
v is the velocity of the particle
\[a=\dfrac{dv}{dt}\]
a is the acceleration of the particle,
\[\dfrac{dv}{dt}\] is the rate of change of velocity

Complete step by step answer:
We know that if the derivative of position is taken we can calculate the speed of the particle. And we have been given position of the particle as a function of time
Let us assume that \[x=3\sin (\pi t)\]
Now, differentiating x with respect to t
We get,
\[\dfrac{dx}{dt}=\dfrac{d(3\sin (\pi t))}{dt}\]
We know first derivative of position=n is velocity
Therefore,
\[v=3\cos (\pi t)\]
We have been asked to calculate velocity of particle at t = 1 second
After substituting
We get,
\[v=3\cos (\pi )\] …………….. (1)
Now, we know that cos\[\pi \] = -1
Therefore,
v = -3
Now, for acceleration we know that the derivative of v is the acceleration
Therefore
\[a=\dfrac{dv}{dt}\]
From (1) we get
\[\dfrac{dv}{dt}=\dfrac{d(3\cos (\pi t))}{dt}\]
We know that derivative of cos is –sin
Therefore,
\[a=-3\sin (\pi t)\]
Again, we have been asked to calculate the acceleration at t = 1 second
Therefore,
\[a=-3\sin (\pi )\]
We know sin\[\pi \] = 0
Therefore,
a = 0
Therefore, the speed of the given particle at t = 1 sec is -3 and acceleration is 0.

Note:
We have not specified the units in our answer as, in the question we have not been given units for the position. We know that, derivative of position function is the instantaneous velocity and the derivative of velocity is the instantaneous acceleration. Therefore, if the acceleration is zero we can say that at the instant the particle is moving with a constant velocity.