$A$ block is placed on a rough horizontal plane. $A$ time-dependent horizontal force $F = Kt$ acts on the block,where $K$ is a positive constant. The acceleration-time graph of the block is:

$A$ spring is compressed between two toy carts of masses $m_1$ and $m_2$. When the toy carts are released,the spring exerts equal and opposite forces for the same time $t$ on each toy cart. If the coefficients of friction $\mu$ between the ground and the toy carts are equal,then the ratio of the displacements of the toy carts is:

$A$ $1.0 \ kg$ block of wood sits on top of an identical block of wood,which sits on top of a flat level table made of plastic. The coefficient of static friction between the wood surfaces is $\mu_1$,and the coefficient of static friction between the wood and plastic is $\mu_2$. $A$ horizontal force $F$ is applied to the top block only,and this force is increased until the top block starts to move. The bottom block will move with the top block if and only if

$A$ block of mass $M = 10\,kg$ rests on a horizontal table. The coefficient of friction between the block and the table is $\mu = 0.05.$ When hit by a bullet of mass $m = 50\,g$ moving with speed $v,$ which gets embedded in it,the block moves and comes to a stop after moving a distance of $2\,m$ on the table. If a freely falling object were to acquire speed $\frac{v}{10}$ after being dropped from height $H,$ then neglecting energy losses and taking $g = 10\,m/s^2,$ the value of $H$ is close to ................. $km$.

At the moment $t=0$, a time-dependent force $F=at$ (where $a$ is a constant equal to $1 \,Ns^{-1}$) is applied to a body of mass $1 \,kg$ resting on a smooth horizontal plane as shown in the figure. If the direction of this force makes an angle $45^{\circ}$ with the horizontal, then the velocity of the body at the moment it leaves the plane is (acceleration due to gravity $=10 \,ms^{-2}$)

Given $m_1 = 4m_2$. How much extra distance (in $cm$) does $m_2$ travel before coming to rest?