**Key concepts:**

1. If two objects are moving in opposite directions towards each other at speeds $u$ and $v$, then

Relative speed = Speed of first + Speed of second = $u + v.$

2. If the two objects move in the same direction with speeds u and v, then

Relative speed = difference of their speeds = $u - v$.

**Important Models:**

**Model 1.**one Pole and one Train:

Length of The Train (m) = Speed of the Train (m/s) × Time taken to cross the pole (s)

Formula: $L = v \times t$

**Model 2.**one Train and one Bridge:

Length of the Train + Length of the Bridge (m) = Speed of the Train (m/s) × Time taken to cross the bridge(s)

Formula: ${L_1} + {L_2} = v \times t$

**Model 3**. one Train with speed speed $u$ and one moving person with speed $v$

**Case 1:**If both are moving in same direction

Length of The Train (m) = [Speed of the Train - Speed of the Man] (m/s) × Time taken to cross the man (s)

Formula: $L = \left( {u - v} \right) \times t$

**Case 2:**If both are moving in opposite direction

Length of The Train (m) = (Speed of the Train + Speed of the Man) (m/s) × Time taken to cross the man (s)

Formula: $L = \left( {u + v} \right) \times t$

**Model 4.**2 Trains with speeds $v$, $u$

**Case 1:**If both are moving in same direction

(Length of The Train 1 + Length of the Train 2)(m) = (Speed of the Train1 - Speed of the Train 2) (m/s) × Time taken to cross (s)

Formula: ${L_1} + {L_2} = \left( {u - v} \right) \times t$

**Case 2:**If both are moving in opposite direction

(Length of The Train 1 + Length of the Train 2)(m) = (Speed of the Train1 + Speed of the Train 2) (m/s) × Time taken to cross (s)

Formula: ${L_1} + {L_2} = \left( {u + v} \right) \times t$