A question about Rectilinear motion

[Priyadarshi Raj]

Homework Statement

Q:
If a particle moving along a straight line under uniform acceleration covers successive equal distances ( S each) in time intervals t₁, t₂ and t₃ respectively, then the expression for average speed of particle in covering the given distance of 3S is:

Options:
A) (S/t₁) + (S/t₂) + (S/t₃)
B) (S/t₁) - (S/t₂) + (S/t₃)
C) 3S / (t₁ + t₂ + t₃)
D) (S/t₁) + (S/t₂) - (S/t₃)

[You may assume the particle is speeding up for the entire journey]

Here the correct answer are options B and C.
I cannot see how B is correct, although I easily got the option C.

Homework Equations

Average speed = Total distance covered / Total time taken

The Attempt at a Solution

Avg speed = (S + S + S) / (t1 + t2 +t3) = 3S / (t1+t2+t3)
This gives option C.
But I cannot get option B anyhow.

 

[Doc Al]

Avg speed = (S + S + S) / (t1 + t2 +t3) = 3S / (t1+t2+t3)
This gives option C.

Good.

But I cannot get option B anyhow.

Hint: Each term in B is an average speed. Express those averages in terms of initial and final speed.

 

[Priyadarshi Raj]

Hint: Each term in B is an average speed. Express those averages in terms of initial and final speed.

Okay then, I believe this is the way:
v₂ will be average of v₁ and v₃
If the avg speeds are v₁ = (S/t₁) , v₂ = (S/t₂), v₃ = (S/t₃), then
Distance traveled during t₂ is given by
S = v₁t₂ + 0.5at₂² and also a = (v₃ - v₁)/t₂ [using Newton's laws of motion]
∴ S = v₁t₂ + 0.5(v₃ - v₁)t₂
⇒ (S/t₂) = v₁ + 0.5v₃ - 0.5v₁
⇒ 2(S/t₂) = v₁ + v₃
⇒ 2(S/t₂) = (S/t₁) + (S/t₃)
⇒ (S/t₂) = (S/t₁) - (S/t₂) + (S/t₃) = v₂
⇒ v_avg = v₂ = (S/t₁) - (S/t₂) + (S/t₃)

Am I correct?

 

[Doc Al]

Okay then, I believe this is the way:
v2 will be average of v1 and v3
If the avg speeds are v1 = (S/t1) , v2 = (S/t2), v3 = (S/t3), then

I'm not sure I follow your reasoning.

Here's how I would look at it. For interval t₁, let the initial and final speeds be v₁ and v₂. Express the average speed (S/t₁) in terms of v₁ and v₂. Do the same for the other intervals and the total.

 

[Priyadarshi Raj]

I'm not sure I follow your reasoning.

Here's how I would look at it. For interval t₁, let the initial and final speeds be v₁ and v₂. Express the average speed (S/t₁) in terms of v₁ and v₂. Do the same for the other intervals and the total.

Thank you for helping me.
The problem is solved.