Problem $1$

NASA is launching a rocket for the Mars mission. At the time of launch, the rocket has a total mass of $\backslash ($ M$=2\mathrm{.}8\backslash$times $10^\{6\}\backslash$mathrm$\{$~kg$\}\backslash ),$ which includes $\backslash (2\mathrm{.}0\backslash$times $10^\{6\}\backslash$mathrm$\{$~kg$\}\backslash )$ of fuel. To lift the rocket, the engines expel exhaust gases backward with an exhaust speed of $\backslash ($ v$\_\{\backslash$mathrm$\{$ex$\}\}=2500\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )$ relative to the rocket.

The fuel is ejected at a rate of $\backslash (1\mathrm{.}4\backslash$times $10^\{4\}\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$s$\}\backslash ),$ providing the necessary thrust to propel the rocket upward.

a$.$ Find the thrust of the rocket

b$.$ Ignore air resistance, calculate the initial acceleration at launch time.

Problem $2$

In the forensic analysis lab, a Morgan State University law enforcement officer is testing the speed of several bullets fired at various targets for accuracy. As shown in the diagram, a bullet of mass $10$ grams is shot into a suspended block of wood of mass of $2$ kg $.$ When the bullet hits the block, it is embedded in the block thus causing the block to swing upward, reaching a height of $5$ cm $.$

A$.$ Using the conservation of linear momentum and conservation of mechanical energy of the system, determine the initial speed of the bullet.

$[6$ points$]$

B$.$ Determine the change in the kinetic energy of the system resulting from the collision.

$[3$ points$]$

Problem $3$

A construction student is working on the remodeling of the white house for the incoming administration testing the effectiveness of a hammer in driving nails into wood. The student swung the hammer weighing approximately $0\mathrm{.}450$ kg horizontally at a speed of $\backslash (7\mathrm{.}00\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ).$ When it hits the nail, it came to a complete stop after pushing the nail $1\mathrm{.}00$ cm into a wooden board.

A$.$ Calculate the time it took for the hammer to bring the nail to a stop. $[4$ points$]$

B$.$ Applying Newton's second law, find the average force applied by the hammer to drive the nail into the wood.

$[2$ points$]$

Problem $4$

A $\backslash (5\mathrm{.}50-\backslash$mathrm$\{$kg$\}\backslash )$ bowling ball moving at $\backslash (9\mathrm{.}00\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )$ collides with a $\backslash (0\mathrm{.}850-\backslash$mathrm$\{$kg$\}\backslash )$ bowling pin, which is scattered at an angle of $\backslash (85\mathrm{.}0^\{\backslash$circ$\}\backslash )$ to the initial direction of the bowling ball and with a speed of $\backslash (15\mathrm{.}0\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ).$

A$.$ Calculate the final velocity $($magnitude and direction$)$ of the bowling ball. $[6$ points$]$

B$.$ From your solution in part A$,$ explain the type of collision in this system.