MHTCET 14 May 2023 Shift 2 Solved Paper

To find when the acceleration of the particle is zero, we first need to understand how acceleration relates to position mathematically. Acceleration is the second derivative of position with respect to time. Given the position function
x=at2−bt3,
let's first find the velocity of the particle, which is the first derivative of position with respect to time (‌

dx

dt

), and then the acceleration, which is the second derivative of position with respect to time (‌

d2x

dt2

).
The first derivative (velocity) is:
v=‌

dx

dt

=‌

d

dt

(at2−bt3)=2at−3bt2
Now, let's take the second derivative (acceleration):
a(t)=‌

d2x

dt2

=‌

d

dt

(2at−3bt2)=2a−6bt.
To find when the acceleration is zero, set the acceleration function to zero and solve for t :
2a−6bt=0.
We can solve for t as follows:
2a‌=6bt
t‌=‌

2a

6b

t‌=‌

a

3b

Therefore, the acceleration of the particle will be zero at t=‌

a

3b

.
This corresponds to option C, which is the correct answer.