If !!m_1 < m_2!! and the floor is smooth. If the car accelerates towards right:
- Rope will be under tension
- Rope will be slack
There is friction between !!m_1!! and !!m_2!! which helps in accelerating.
We can solve it using 2 methods.
First suppose the rope is under tension. Let !!T!! be the tension in the string, !!F!! be the friction between !!m_1!! and !!m_2!!
So on !!m_1!!:
So !!T= (m_1-m_2)a!!
But !!m_1 < m_2!! so !!T<0!! which is not possible since it is rope.
Since the same frictional force is acting on !!m_1,m_2!!. But !!m_1 < m_2!! and the same tension is acting on !!m_1,m_2!!. So the acceleration of !!m_1 > m_2!!, hence the rope will be slack and there will be no tension in the rope.