$A$ ball of mass $m=1 \, kg$ is thrown from the top of a building with initial velocity $v=(20 \, m/s) \hat{i} + (24 \, m/s) \hat{j}$ at time $t=0$. The change in the potential energy of the ball between $t=0$ and $t=6 \, s$, if the ball does not hit the ground, is (assume $g=10 \, m/s^2$): (in $ \, J$)

$A$ ball is rolled off the edge of a horizontal table at a speed of $4\, m/s$. It hits the ground after $0.4\, s$. Which statement given below is true?

$A$ bullet must be fired from a height of ........ $cm$ to hit a target $100 \,m$ away. The horizontal velocity of the bullet is $500 \,ms^{-1}$. (Take $g = 10 \,ms^{-2}$)

$A$ fighter plane is flying horizontally at a height of $250\ m$ from the ground with a constant velocity of $500\ m/s$. It passes exactly over a cannon which can fire a shell at any time in any direction with a speed of $100\ m/s$. Find the duration of time for which the plane is in danger of being hit by a cannon shell.

$A$ man runs across the roof of a tall building and jumps horizontally with the hope of landing on the roof of the next building,which is at a lower height than the first. If his speed is $9 \, m/s$,the horizontal distance between the two buildings is $10 \, m$,and the height difference is $9 \, m$,will he be able to land on the next building? (Take $g = 10 \, m/s^2$)

$A$ helicopter flying horizontally with a speed of $360 \ km/h$ at an altitude of $2 \ km$,drops an object at an instant. The object hits the ground at a point $O$,$20 \ s$ after it is dropped. The displacement of $O$ from the position of the helicopter where the object was released is: (use acceleration due to gravity $g = 10 \ m/s^2$ and neglect air resistance)