Problem #$1$:

The $6-$lb particle is subjected to the action of its weight and the following forces:

$F_1=\{2i+6j-2tk\}lb$

$F_2=\{t^{2}i-4tj-1k\}lb,$ and

$F_3=\{-2ti\}lb,$ where $t$ is the time in seconds.

Determine the distance the ball is from the origin $2$ s after being released from rest.

Problem #$2$:

If the $50-kg$ crate starts from rest and achieves a velocity of $v=4\frac{m}{s}$ when it travels a distance of $5$ m to the right. determine the magnitude of force P acting on the crate. The coefficient of kinetic friction between the crate and the ground is ${}_k=0\mathrm{.}3.$

Problem #$3$:

The $50-kg$ crate shown rests on a horizontal surface for which the coefficient of kinetic friction is ${}_k=0\mathrm{.}3.$ If the crate is subjected to a $400-N$ towing force as shown, determine the velocity of the crate in $3$ s starting from rest.

Problem #$4$:

The $100-kg$ block A shown is released from rest. If the masses of the pulleys and the cord are neglected, determine the velocity of the $20-kg$ block B in $2$s$.$