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(Solved):   Two particles of masses \( m_{1} \) and \( m_{2} \) collide on a frictionless surface, see ...



Two particles of masses \( m_{1} \) and \( m_{2} \) collide on a frictionless surface, see Figure 1 . Particle \( m_{1} \) ha\( F_{1} \)

 

Two particles of masses \( m_{1} \) and \( m_{2} \) collide on a frictionless surface, see Figure 1 . Particle \( m_{1} \) has an initial speed of \( u_{1} \) in the x-direction and particle \( m_{2} \) is initially stationary. After the collision particle \( m_{1} \) has a speed \( v_{1} \) with a trajectory that makes an angle \( \theta_{1} \) with the \( \mathrm{x} \)-axis, and particle \( m_{2} \) has a speed \( v_{2} \) with a trajectory that makes an angle \( \theta_{2} \) with the \( \mathrm{x} \)-axis. a) Determine the following kinematic qualities of the system in terms of \( v_{1}, v_{2}, \theta_{1} \) and \( \theta_{2} \). make sure to show your work by drawing the appropriate triangles. a. \( v_{1, x}= \) b. \( v_{1, y}= \) c. \( v_{2, x}= \) d. \( v_{2, y}= \) b) Write the equations for conservation of momentum in the \( x \) and \( y \) directions, a. x-direction: b. y-direction: c) Using the y-direction equation you derived above, a) Solve for the final speed of particle \( 1, v_{1} \), in terms of \( m_{1}, m_{2}, \theta_{1}, \theta_{2} \) and \( v_{2} \). (b) Using the equation from (a) determine the relationship between \( m_{1}, m_{2}, \sin \left(\theta_{1}\right) \), and \( \sin \left(\theta_{2}\right) \) that must be true if the final speeds of both particles are equal (i.e. \( \left.v_{1}=v_{1}\right) \). a. \( v_{1}= \) b. \( \frac{m_{1}}{m_{2}}= \) d) Using the equation for \( v_{1} \) derived in part (c), determine the equation for the final speed of particle \( 2, v_{2} \), in terms of \( m_{1}, m_{2}, \theta_{1}, \theta_{2} \) and \( u_{1} \). a. \( v_{2}= \) e) Use the equation for \( v_{2} \) derived in part d) to determine the equation for the final speed of particle \( 1, v_{1} \), in terms of \( m_{1}, m_{2}, \theta_{1}, \theta_{2} \) and \( u_{1} \). a. \( v_{1}= \) f) Using the following values: i. \( u_{1}=10 \frac{\mathrm{m}}{\mathrm{s}} \) ii. \( \theta_{1}=45^{\circ} \) iii. \( \theta_{2}=30^{\circ} \) iv. \( m_{2}=5 \mathrm{~kg} \) Determine the value of \( m_{1} \) that will make the collision elastic: \( m_{1}= \) \( F_{1} \)


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(a) From the triangle ?ABC : AB=v1,x=v1cos??1 CB=v1,y=v1sin??1 From the triangle ?AFD : AF=v2,x=v2cos??2 FD=v2,y=?v2sin??2 The following image provide
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