WBJEE 2024 Math Paper

To find the maximum velocity and the corresponding time when the particle reaches this velocity, we need to analyze the given acceleration function f, which is:
f=6−√1.2t

Step 1: Finding the time T when the acceleration becomes zero
The velocity of the particle will be maximum when the acceleration becomes zero, because acceleration describes the rate of change of velocity. Therefore, we set the acceleration function to zero and solve for t.

6−√1.2t=0
Solving for t :
‌√1.2t=6
‌1.2t=36
‌t=‌

36

1.2

‌t=30 sec

So, the time T when the acceleration becomes zero and the velocity is at its maximum is 30 seconds. Therefore, Option C is correct.
Step 2: Finding the maximum velocity v
The velocity can be found by integrating the acceleration function with respect to time. Since the particle starts from rest, the initial velocity v(0) is zero.
The velocity function v(t) is given by:
v(t)=∫(6−√1.2t)‌dt
We can solve this integral by splitting it into two parts:
v(t)=∫6‌dt−∫√1.2t‌dt
Now, calculate the integral step-by-step:
∫6‌dt=6t

For the second integral, let u=1.2t. Then, du=1.2‌dt, or dt=‌

du

1.2

.
Therefore, the velocity function is:
v(t)=6t−‌

5

9

(1.2t)‌

3

2

Now, substitute t=30 sec to find the maximum velocity:
‌v(30)=6⋅30−‌

5

9

(1.2⋅30)‌

3

2

‌=180−‌

5

9

(36)‌

3

2

‌=180−‌

5

9

⋅216
‌=180−120
‌=60‌ft∕ sec

Therefore, the maximum velocity v is 60‌ft∕ sec. Thus, Option B is correct.
In conclusion, the correct answers are:
Option B: v=60‌ft∕ sec
Option C: T=30 sec