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Back to 8.6 Collisions of Point Masses in Two Dimensions

8.6 Exercise 10

Avg difficulty 2.5

Starting with equations m1v1=m1 v1′cosθ1+m2 v2′cosθ2m_{1} v_{1} = m_{1} v_{1}^{'} \text{cos} \theta_{1} + m_{2} v_{2}^{'} \text{cos} \theta_{2} and 0=m1 v1′sinθ1+m2 v2′sinθ20 = m_{1} v_{1}^{'} \text{sin} \theta_{1} + m_{2} v_{2}^{'} \text{sin} \theta_{2} for conservation of momentum in the xx- and yy-directions and assuming that one object is originally stationary, prove that for an elastic collision of two objects of equal masses,

1 2 mv 1 2 = 1 2 mv ′ 1 2 + 1 2 mv ′ 2 2 + mv′ 1 v ′ 2 cos ( θ 1 − θ 2 ) \begin{matrix}\frac{1}{2} \left(\text{mv}_{1}\right)^{2} = \\ \frac{1}{2} \left(\left(\text{mv} '\right)_{1}\right)^{2} + \frac{1}{2} \left(\left(\text{mv} '\right)_{2}\right)^{2} \\ + \left(\text{mv} '\right)_{1} \left(v '\right)_{2} \text{cos} \left(\theta_{1} - \theta_{2}\right)\end{matrix}

as discussed in the text.

OpenStax College Physics 2e, 8.6 Collisions of Point Masses in Two Dimensions, exercise 10

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Adapted from College Physics © 2022 Rice University. Licensed under CC BY 4.0. Sources: College Physics 2e, College Physics for AP® Courses 2e. Imported at revision f98d7a7.