Question

Two cars start off to race with v1 and v2 travel in a straight line with uniform acceleration a1 and a2 respectively. If the race ends in a dead heat (I.e they reach the finishing point at the same time) prove the length of the course is 2 (v1-v2)(v1av-v2a2)(a1-a2)/(a1-a2)^2

Answer 1

Let distance covered by cars = L and time taken is t
For first car :-
initial velocity = v₁
Acceleration = a₁
Time taken = t
∴ use formula S = ut + 1/2at²
L = v₁t + 1/2 a₁t² --------(1)

For 2nd car :-
initial velocity = v₂
acceleration = a₂
time taken = t
so, L = v₂t + 1/2 a₂t² -----(2)

From equations (1) and (2),
v₁t + 1/2a₁t² = v₂t + 1/2a₂t²
⇒(v₁ - v₂)t = 1/2(a₂ - a₁)t²
⇒t = 2(v₁ - v₂)/(a₂ - a₁) , put it in equation (1)

L = v₁[2(v₁ - v₂)/(a₂ - a₁)] + 1/2a₁[2(v₁ - v₂)/(a₂ - a₁)]²
= (v₁ - v₂)/(a₂ - a₁)[2v₁ + 2a₁(v₁ - v₂)/(a₂ - a₁)]
= (v₁ - v₂)/(a₂ - a₁) [2v₁(a₂ - a₁) + 2a₁(v₁ - v₂)]/(a₂ - a₁)
= (v₁ - v₂)/(a₂ - a₁)² [2v₁a₂ - 2v₁a₁ + 2v₁a₁ -2a₁v₂]
= 2(v₁ - v₂)(v₁a₂ - a₁v₂)/(a₂ - a₁)²

Hence proved //

[Note :- you did little mistake in typing . See my answer ]

Answer 2

Hello Dear.

To Prove ⇒ 2(v₁ - v₂)(v₁a₂ - a₁v₂)/(a₂ - a₁)² 

[There is the little typing error in the Question. We can prove the above written Equation. It is correct.]


Proof ⇒

Let the distance covered by the car be x. & the time taken by the car be t.
[since, they reach the finishing point at the same time).

Now, For the First Car,
Acceleration = a₁
Initial Velocity = v₁
Time taken by the car to reach the Finishing point = t sec.

According to the second equation of the motion,

S = ut + 1/2 at²

where . S = Distance covered by the car, u = initial velocity, a = acceleration, and t = time taken.

∴ x = v₁ × t + 1/2 × a₁ × t²
⇒ x = v₁t + a₁t²/2

For Second Car,
Acceleration = a₂
Initial Velocity = v₂
Time taken = t

∴ S = v₂ × t + 1/2 × a₂ × t²
⇒ x = v₂ + a₂ t²/2

Now we have,
x = x
v₁t + a₁t²/t = v₂t + a₂ t²/2
⇒ v₁t - v₂t = a₂t²/2 - a₁t²/2
⇒ t(v₁ - v₂) = t²/2(a₂ - a₁)
⇒ t = 2(v₁ - v₂) ÷ (a₂ - a₁)

Now, Putting this value in the Equation distance covered by the First car.

∴ x = v₁[2(v₁ - v₂)/(a₂ - a₁)] + 1/2 a₁ {2(v₁ - v₂)/(a₂ - a₁)}² 
 x = [(v₁ - v₂)/(a₂ - a₁)] {2v₁ + 2a₁(v₁ - v₂)/(a₂ - a₁)} 
 x = [(v₁ - v₂)/(a₂ - a₁)] {2v₁(a₂ - a₁) + 2a₁(v₁ - v₂)}/(a₂ - a₁)
x = (v₁ - v₂)/(a₂ - a₁)² [2v₁a₂ - 2v₁a₁ + 2v₁a₁ -2a₁v₂]

x = 2(v₁ - v₂)(v₁a₂ - a₁v₂)/(a₂ - a₁)² 


Hence Proved.

Hope it helps.