The Question is from neet-physics motion-in-a-straight-line acceleration-and-horizontal-motion, and it was created by the author SUBHANI

Question

A particle starts from rest and moves along a straight line with an acceleration given by $a = 6t - 4$, where $a$ is in $m/s^2$ and $t$ is in seconds.  At what time is the velocity of the particle zero after the start of motion?

SUBHANI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: B
## CONCEPT FRAMEWORK
	**Primary Concept**: Kinematics
	**Related Topics**: Integration, Velocity, Acceleration
	**Prerequisites**: Basic calculus, understanding of acceleration as the derivative of velocity
## SOLUTION PATHWAY
**Given Information**:
	* Acceleration, a(t) = 6t - 4 m/s^2
	* Initial velocity, v(0) = 0 m/s (starts from rest)
	* Required: Time 't' when velocity, v(t) = 0 m/s
**Method & Steps**:
	1.	Find the velocity function v(t) by integrating the acceleration function a(t).
		Reasoning: Velocity is the integral of acceleration with respect to time.
		Formula/Rule: $v(t) = \int a(t) dt$
	2.	Perform the integration and apply the initial condition to find the constant of integration.
		Working: $v(t) = \int (6t - 4) dt = 3t^2 - 4t + C$. Since the particle starts from rest, v(0) = 0, so $0 = 3(0)^2 - 4(0) + C$, which means C = 0. Therefore, $v(t) = 3t^2 - 4t$.
		Key Point: The initial condition is crucial to find the constant of integration.
	3.	Set the velocity function v(t) equal to zero and solve for t.
		Result: $0 = 3t^2 - 4t$. Factoring out 't' gives $t(3t - 4) = 0$. Thus, t = 0 or $3t - 4 = 0$.  Solving $3t - 4 = 0$ gives $t = 4/3$ seconds. Since we want the time after the start, t = 0 is not the answer, so the answer is 4/3 seconds.
		Verification: Substituting t = 4/3 back into the velocity equation gives $v(4/3) = 3(4/3)^2 - 4(4/3) = 3(16/9) - 16/3 = 16/3 - 16/3 = 0$.
 ## OPTION ANALYSIS
[A] :x: 2/3 seconds. This is not the correct time calculated when the velocity is zero after the start of motion. 
[B] ✓: 4/3 seconds. This is the correct time when the velocity is zero after the start of motion as calculated from integration and solving for velocity zero. 
[C] :x: 1 second. This is not the correct time calculated when the velocity is zero after the start of motion.
[D] :x: 3/2 seconds. This is not the correct time calculated when the velocity is zero after the start of motion.
 ## LEARNING AIDS
	**Common Mistakes**:
		1.[Error 1]: Forgetting to include the constant of integration when integrating. This can lead to an incorrect velocity function.
		2.[Error 2]: Not setting the velocity equal to zero to find the time when velocity is zero. This is the key step to solving the problem.
	**Quick Tips**:
		1.[Helpful shortcut]: Remember that integration of acceleration gives velocity, and that initial conditions are used to find the constant of integration.
		2.[Memory aid]: Think of the process as 'acceleration to velocity' (integration), then 'velocity to time' (solving v(t) = 0).

SUBHANI · Answer

2/3 seconds

SUBHANI · Answer

The correct Option is "4/3 seconds"

SUBHANI · Answer

1 second

SUBHANI · Answer

3/2 seconds

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