The Question is from neet-physics motion-in-a-straight-line , and it was created by the author Mr. ANIL KUMAR.G

Question

A particle moving with uniform acceleration along a straight line covers distances $S_1$ and $S_2$ in successive time intervals $t_1$ and $t_2$.  Which equation correctly relates these quantities?

Mr. ANIL KUMAR.G · Accepted Answer

## CORE INFORMATION
✓ **Answer**: B
## CONCEPT FRAMEWORK
	**Primary Concept**: Kinematics of Uniformly Accelerated Motion
	**Related Topics**: Equations of motion, displacement, time, acceleration
	**Prerequisites**: Understanding of displacement, velocity, acceleration, and basic algebraic manipulation.
## SOLUTION PATHWAY
**Given Information**:
	* Displacement $S_1$ in time $t_1$
	* Displacement $S_2$ in time $t_2$
	* Uniform acceleration '$a$'
**Method & Steps**:
	1.	Express the displacement in terms of initial velocity and acceleration for the first interval.
		Reasoning: We use the equation of motion: $s = ut + (1/2)at^2$
		Formula/Rule: $S_1 = u t_1 + (1/2) a t_1^2$
	2.	Express the displacement in terms of initial velocity (at start of t1) , acceleration and total time for the combined interval.
		Reasoning: The displacement for the combined interval is $S_1 + S_2$ and time is $t_1 + t_2$. We can use the same equation of motion.
		Working: $S_1 + S_2 = u(t_1 + t_2) + (1/2)a(t_1+t_2)^2$
	3.	Express the displacement $S_2$ from the end of time $t_1$ to the end of $t_2$. Let v be the velocity at the start of $t_2$.  $v = u+at_1$.  
		Reasoning: We can use the same equation of motion for the second interval $S_2$ with the initial velocity as $v$ at the start of $t_2$.
		Formula/Rule: $S_2 = v t_2 + (1/2) a t_2^2 = (u+at_1)t_2 + (1/2)at_2^2= ut_2+at_1t_2 + (1/2)at_2^2$
	4.	Solve for $u$ from $S_1$ and sub into $S_2$ equation.
		Working: $u = (S_1 - (1/2)at_1^2)/t_1$.  $S_2 =  (S_1 - (1/2)at_1^2)/t_1*t_2+at_1t_2 + (1/2)at_2^2$.   $S_2 =  (S_1t_2)/t_1 - (1/2)at_1t_2+at_1t_2 + (1/2)at_2^2$ 
		Key Point: Simplify $S_2 = (S_1t_2)/t_1 + (1/2)at_1t_2 + (1/2)at_2^2$ and $S_2/t_2= S_1/t_1 +(1/2)at_1+ (1/2)at_2$ so $S_2/t_2 - S_1/t_1 = (1/2)a(t_1+t_2)$.
	5.	Rearrange to get the final equation
		Result: $S_2/t_2 - S_1/t_1 = a(t_1+t_2)/2$
		Verification: The derived equation matches option B.
 ## OPTION ANALYSIS
[A] :x: Incorrect relation of variables.
[B] ✓: Correctly relates the given quantities based on the equations of motion.
[C] :x: Incorrect relation of variables.
[D] :x: Incorrect relation of variables.
 ## LEARNING AIDS
	**Common Mistakes**:
		1.[Error 1]: Incorrectly applying the equations of motion, mixing up initial and final velocities or displacements. [How to avoid]: Carefully define the start and end points for each interval and use the appropriate kinematic equation.
		2.[Error 2]: Algebraic errors in simplification and solving for the unknowns. [How to avoid]: Double-check each step of your algebraic manipulation.
	**Quick Tips**:
		1.[Helpful shortcut]: When dealing with successive intervals, try to express the initial velocity for the second interval in terms of the first interval's final velocity.
		2.[Memory aid]: Remember the basic equations of motion: $s = ut + (1/2)at^2$, $v = u + at$, and $v^2 = u^2 + 2as$ and choose the one that fits the problem.

Mr. ANIL KUMAR.G · Answer

S_2t_1 - S_1t_2 = a(t_1+t_2)

Mr. ANIL KUMAR.G · Answer

The correct Option is "S_2/t_2 - S_1/t_1 = a(t_1+t_2)/2"

Mr. ANIL KUMAR.G · Answer

S_2t_1 + S_1t_2 = a(t_1+t_2)

Mr. ANIL KUMAR.G · Answer

S_2/t_1 - S_1/t_2 = a(t_1+t_2)/2

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