Consider a particle moving along the x-axis where x(t) is the position of the particle at time t, x'(t) is its velocity, and x"(t) is its acceleration

Jason Farmer 2020-11-10 Answered
Consider a particle moving along the x-axis where x(t) is the position of the particle at time t, x'(t) is its velocity, and x"(t) is its acceleration.
x(t)=t36t2+9t9,
0t10
a) Find the velocity and acceleration of the particle?
x(t)=?
x"(t)=?
b) Find the open t-intervals on which the particle is moving to the right?
c) Find the velocity of the particle when the acceleration is 0?
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Expert Answer

Clelioo
Answered 2020-11-11 Author has 88 answers

Step 1
Position as a function of time is given. We have to find the velocity and acceleration.
Recall the famous rule of differentiation:
d(xn)/dx=nxn1
Part (a)
Velocity =x(t)=3t212t+9
Acceleration =x"(t)+6t12
Part (b)
When particle is moving to the right, velocity >0
Hence, x(t)>0
Hence, 3t212t+9>0
Or, t24t+3>0
Or, (t3)(t1)>0
Hence, t<1 or t>3
The bounds for t as stated in question is: 0t10
Hence, the open t-interval will be: [0,1)(3,10]
Part (c)
Acceleration is zero when x"(t)=6t12=0
Hence, t=12/6=2
Hence, velocity when acceleration is zero =x(t=2)=322122+9=3

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