Saurabh Koranglekar
Last Activity: 6 Years ago
To determine the velocity of the particle at the moment when the acceleration becomes zero, we first need to analyze the given acceleration function, which is defined as f = fo(1 - t/T). Here, fo is a constant representing the initial acceleration, and T is another constant related to the time scale of the system. We start by finding the time at which the acceleration becomes zero.
Finding the Time When Acceleration is Zero
Setting the acceleration function to zero gives us:
0 = fo(1 - t/T)
Solving for t yields:
1 - t/T = 0
t = T
Understanding Velocity Through Integration
Since we know the particle starts with zero velocity at t = 0, we can find the velocity function by integrating the acceleration function with respect to time. The acceleration f can be expressed as:
a(t) = fo(1 - t/T)
Now, we can integrate to find the velocity v(t):
Integrating the Acceleration
The velocity is the integral of acceleration:
v(t) = ∫ a(t) dt = ∫ fo(1 - t/T) dt
Carrying out the integration:
v(t) = fo [t - (t^2)/(2T)] + C
Since the initial velocity v(0) = 0, we can determine C:
v(0) = fo [0 - 0] + C = 0 ⇒ C = 0
Thus, the velocity function simplifies to:
v(t) = fo [t - (t^2)/(2T)]
Calculating Velocity at t = T
Now, we substitute t = T to find the velocity when the acceleration is zero:
v(T) = fo [T - (T^2)/(2T)]
v(T) = fo [T - T/2] = fo (T/2)
Final Result
This means that the velocity of the particle at the instant when the acceleration becomes zero is:
v(T) = (1/2) fo T
Conclusion and Options
From the options provided:
- 1) 1/2 fo T^2
- 2) fo T^2
- 3) (1/2) fo T
- 4) fo T
The correct answer is option 3: (1/2) fo T.
This step-by-step analysis not only reveals how we arrived at the solution but also highlights the importance of understanding acceleration and velocity relationships in motion along the x-axis.
