A bullet fired into a target losses one-third of its velocity after travelling a distance $x$ metre into the target. If the bullet comes to rest by travelling a further distance $x^{\prime}$, then the ratio $\frac{x^{\prime}}{x}$ is

An ant starts from the origin and crawls 10 cm along the $X$-axis and then 20 cm along the $Y$-axis. The dot product of the ant's displacement vector with the position vector of a point that makes $45^{\circ}$ with the $X$-axis and has a magnitude of $\sqrt{2} \mathrm{~cm}$ is

A projectile is launched with an initial speed of $40 \mathrm{~m} / \mathrm{s}$ at an angle $30^{\circ}$ above the ground. The projectile lands on a hillside 2.0 s later. The net displacement from where the projectile lands on hillside 2.0 s later. The net displacement from where the projectile was launched to where it hits the target is (take, $g=10 \mathrm{~m} / \mathrm{s}^2$ )

Two blocks of masses 1 kg and 2 kg connected by a light rod and the system is slipping down a rough incline angle $45^{\circ}$ with the horizontal. The frictional coefficient at both the contacts is 0.4 . If the acceleration of the system is $\alpha \sqrt{2}$, the value of $\alpha$ is (use, $g=10 \mathrm{~m} / \mathrm{s}^2$ )