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Problem 27 Medium Difficulty

Two ice skaters, Daniel (mass 65.0 $\mathrm{kg} )$ and Rebecca (mass $45.0 \mathrm{kg} ),$ are practicing. Daniel stops to tie his shoelace and, while at rest, is struck by Rebecca, who is moving at 13.0 $\mathrm{m} / \mathrm{s}$ before she collides with him. After the collision, Rebecca has a
velocity of magnitude 8.00 $\mathrm{m} / \mathrm{s}$ at an angle of $53.1^{\circ}$ from her initial direction. Both skaters move on the frictionless, horizontal surface of the rink. (a) What are the magnitude and direction of Daniel's velocity after the collision? (b) What is the change in total kinetic energy of the two skaters as a result of the collision?

Answer

$$\begin{array}{l}{\text { (a) } v_{D f}=7.20 \mathrm{m} / \mathrm{s}, 38.0^{\circ} \text { about }x \text {-axis }} \\ {\text { (b) } \Delta K=-678 \mathrm{J}}\end{array}$$

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Video Transcript

height. The given problem marks off. Daniel is given ass. M t is equal to 65.0 kg and mass off. Rebeka is given as M R is equal to 45.0 kg. The four collision initially speech off Rebecca Waas. We I are is equal to 13.0 m per second And as Daniel Waas addressed so his initial speed v I t is equal to zero now this is their initial direction of motion. Let it be replica. Let it be, Daniel Then here This is the horizontal line. Let it be alone X axis Then after the collision, Rebecca goes here with the finally speech we f b were we f d is given as 8.0 meter per second and it is at an angle off 53.1 degree. Suppose this is we f r. Typical Rebecca is going with finally speech off 8 m per second and the final speed off Daniel is missing. Let it be v f the which we have to find and it is supposed at an angle of theater from the horizontal. So the component if we find a component off the finalists. Feed off the car. This is B F. R course, 53.1 degree. And here this is we f our serene 53.1 degree. Similarly, for VfB for the component off the speed or finally, speed of Daniel let be the speed of Daniel along X axis We f b x, we say it And here this is we f b y means the component off. Finally speed of Daniel along y axis. No. First of all, in the first part of the problem, we have to find the magnitude and direction of the final speed off Daniel, for which we use conservation all linear momentum. A long X X is so it becomes for Rebecca. This is M R into V i. R. For Daniel, it is empty in tow. V idee is equal to a long X axis. For Rebecca, this is M R. In tow, V i v f our cause 53.1 degree. And for Daniel, this is M D into V f d. X now plugging in all the known values for m R. This is 45 kg for we. I are This is 13 m per second. But as the ID is zero, so this term will become zero and here after the collision. 45 kg for the mass Off the car we are This is eight into cause sign off 53.1 degree For Daniel, this is 65 kg in tow. We F B X, which is missing so calculating it here it comes out to be 65. We f d x X component off the final speed off Daniel is equal to 45 into 13 comes out to be 585 miners 216.2, which comes out to be 368.85 So finally, the X component horizontal component off the finalist speech off Daniel here comes out to be 5.68 m per second. Now we have to find the vertical component component alone via taxes, for which we use conservation off linear momentum, a long vai access and along by access. There is no initial speed along by access. Before the collision, there was nothing along by access, so this is zero, and after the collision for ripping carcasses, M R. In tow V F are into sign 53.1 degree plus for Daniel empty in tow. V F de vie along by access so plugging in all non values. It comes out to be 45 kg into 8 m per second into sign 53.1 degree plus 65 kg into V F B five here, which is missing no rearranging that terms it becomes mhm. 65 v f B Y is equal to minus 287.886 Or we can say finally, we FBI means the vertical component component along X axis off the finally speed off. Daniel comes out to be minus 4.43 m per second. Here. This negative sign shows that this component off his speech is a long negative. Why access hands? Finally, we can find the net to speed off Daniel after the collision. So if he mark it like this is the horizontal component, we f b X and this is the vertical component we f b y. Then the finally speed after the collision for Daniel is Theis V. F d. An angle theta Soto find this VfB We used the tradition off this we f b x squared plus we f b y square. So putting the values off fdx and we have these wise we're here. This is 5.68 square plus minus 4.43 square. So here comes out Toby 32.26 to 4 plus 19.6249 on adding it becomes 51.8873 So finally this FD comes out, Toby 7.20 m per second means this is the finally speed off. Daniel. After the religion now find its angle theater, we use 10 theater which has given us we f b y divided by we f d x and we will use only the magnitude of this We f divided is 4.43 m per second, divided by 5.68 need a per second and here it comes out of the 017 799 So here becomes theater is equal toe end in verse 0177 99 So finally Tita is equal to 37.95 degree. Or we can say this is 38 degree. So this is the final velocity off Daniela product collision, having a magnitude of 7.20 m per second and at that time, at an angle of 38 degree from the original direction. No. In the second part of the problem, we have to find the change in kinetic energy off the system. And that will be given us the final kinetic energy off both the Daniel Anta Rebecca. So, for the kinetic energy of Daniel, this is half empty into the F D Square Plus for the kind of technology of typical. This is half M R V F r squared, then minus initial kind of energy off M B. The I d Square plus half of em are the I r squared. Now we take this half as a barman out and putting all other values for MD. This is 65 k g 7.20 square less 45 eight square minus as the idea zero. So just the i r will give us 45 into 13 square. So finally it comes out to be uh huh into 3369.6 plus 2880 minus 7605 jewels. So it becomes minus 677.7 Jews. Or we can say approximately. This is minus 678 Jews. So this is the lost in kinda technology off the system. Thank you.