Kinematics

Two identical bodies A and B of equal masses have initial velocities $\vec v_1=4\hat i$ m/s and $\vec v_2=4\hat j$ m/s respectively. The body A has acceleration $\vec a_1=6\hat i+6\hat j$ m/s$^2$ while the acceleration of the other body B is zero. The centre of mass of the two bodies moves in ______ path.

The two projectiles are projected with the same initial velocities at the $15^\circ$ and $30^\circ$ with respect to the horizontal. The ratio of their ranges is $1:x$. The value of $x$ is

Two cars $A$ and $B$ are moving in the same direction along a straight line with speeds 100 km/h and 80 km/h, respectively such that car $A$ is moving ahead of car $B$. A person in car $B$ throws a stone with a speed $v$ so that it hits the car $A$ with a speed of 5 m/s. The value of $v$ is ______ km/h.

If $x$ and $y$ coordinates of a projectile as a function of time ($t$) are given as $24t$ and $43.6t-4.9t^2$, respectively, then the angle (in degrees) made by the projectile with horizontal when $t=2$ s is ______.

A gun mounted on the ground fires bullets in all directions with same speed. The farthest distance the bullets could reach is 6.4 m. The speed of the bullets from the gun is ______ m/s. (take $g=10$ m/s$^2$)

From 18 m height above the ground a ball is dropped from rest. The height above the ground at which the magnitude of velocity equal to the magnitude of acceleration (in the same set of units) due to gravity is ____ m. (Take $g = 10\ \mathrm{m/s^2}$ and neglect the air resistance)

A gas balloon is going up with a constant velocity of 10 m/s. When this balloon reached a height of 75 m, a stone is dropped from it and balloon keeps moving up with the same velocity. The height of the balloon when the stone hits the ground is ______ m. (Take $g=10$ m/s$^2$)

Two identical bodies, projected with the same speed at two different angles cover the same horizontal range $R$. If the time of flight of these bodies are 5 s and 10 s, respectively, then the value of $R$ is ______ m. (Take $g=10$ m/s$^2$)

Two masses of 3.4 kg and 2.5 kg are accelerated from an initial speed of 5 m/s and 12 m/s, respectively. The distances traversed by the masses in the 5$^{th}$ second are 104 m and 129 m, respectively. The ratio of their momenta after 10 s is $\frac{x}{8}$. The value of $x$ is ______.