$A$ ball is dropped from a height of $5 \ m$ onto the floor of an elevator. The ball bounces off the floor. At the time of collision,the elevator is moving upwards with a velocity of $1 \ m/s$. What is the velocity of the ball immediately after the collision in $m/s$?

Two blocks of equal masses are connected with a massless spring of spring constant $k = 2500 \,N/m$ and natural length $10 \,cm$,resting on a frictionless horizontal plane. If a constant horizontal force $F = 10 \,N$ is applied as shown in the figure,find the maximum distance between the blocks. (in $cm$)

$A$ particle of mass $7 \, kg$ moving at $5 \, m/s$ is acted upon by a variable force opposite to its initial direction of motion. The variation of force $F$ with time $t$ is shown in the graph.

Three forces $\vec{F}_1=(2 \hat{i}+4 \hat{j}) \,N$,$\vec{F}_2=(2 \hat{j}-\hat{k}) \,N$,and $\vec{F}_3=(\hat{k}-4 \hat{i}-2 \hat{j}) \,N$ are applied on an object of mass $1 \,kg$ at rest at the origin. The position of the object at $t=2 \,s$ will be:

Imagine the situation in which the given arrangement is placed inside a trolley that can move only in the horizontal direction,as shown in the figure. If the trolley is accelerated horizontally along the positive $x$-axis with $a_0$,and if $a_{min}$ and $a_{max}$ are the minimum and maximum values of $a_0$ for which the blocks remain stationary with respect to the surface,then identify the correct statements.

$A$ block $A$ with mass $100\, kg$ is resting on another block $B$ of mass $200\, kg$. As shown in the figure,a horizontal rope tied to a wall holds block $A$. The coefficient of friction between $A$ and $B$ is $0.2$,while the coefficient of friction between $B$ and the ground is $0.3$. The minimum required force $F$ to start moving block $B$ will be ........ $N$.