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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 $$m_{1}$$ 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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