Question

42. A car, starting from rest, has a constant
acceleration a, for a time interval t, during which
it covers a distances $1. In the next time intervaltz,
the car has a constant retardation a, and comes to
rest after covering a distance s2 in time tz. Which
of the following relations is correct?​

Answer 1

Correct Question :

A car starting from rest has a constant acceleration a₁ for a time interval t₁ during which it covers a distances S₁. In the next time interval t₂ the car has a constant retardation a₂ and comes to rest after covering a distance S₂ in time interval t₂. Find the relation between distance, acceleration and time.

Solution :

⟩ Distance covered by car in case A :

• Initial velocity = zero
• Final velocity = v₁
• Acceleration = a₁
• time = t₁
• Distance = s₁

➝ s₁ = (0 × t₁) + 1/2 (a₁ × t₁²)

➝ s₁ = a₁t₁² / 2 ___ [Eq. 1]

⟩ Final velocity after time t₁ :

➠ v₁ = 0 + a₁t₁

➠ v₁ = a₁t₁

⟩ Final velocity after time t₂ :

• Final velocity = zero
• Initial velocity = a₁t₁
• Time = t₂
• Acceleration = a₂

➠ 0 = a₁t₁ - a₂t₂

➠ a₁t₁ = a₂t₂ ___ [Eq. 2]

➠ t₁ / t₂ = a₂ / a₁ ___ [Eq$.$ 3]

⟩ Distance covered by car in case B :

➝ s₂ = (a₁t₁) t₂ - 1/2 (a₂ × t₂²)

From the second equation;

➝ s₂ = a₂t₂² - 1/2 a₂t₂²

➝ s₂ = 1/2 a₂t₂² ___ [Eq. 4]

From the first and fourth equation;

➙ s₁ / s₂ = (a₁t₁²/2) / (a₂t₂²/2)

From the third equation;

➙ s₁ / s₂ = (a₁ / a₂) × (a₂² / a₁²)

➙ s₁ / s₂ = a₂ / a₁ = t₁ / t₂