[PHYSICS 7A] Homework 07 Solutions

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University of California, Berkeley **We aren't endorsed by this school
Course
PHYSICS 7A
Subject
Electrical Engineering
Date
Oct 9, 2023
Pages
1
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Homework 7: Center of Mass, Momentum and Collisions Physics for Scientists and Engineers, 7A - William Golightly - University of California, Berkeley Question 1 Part A Question 5: Collision at an Angle Part A Since there is no external force on the system, we use the conservation of momentum: p i = p f (1) m ( v cos ϕ ) = 2 mv final sin θ (2) m ( v sin ϕ ) + 2 mv = 2 mv final cos θ (3) From Equations (2) and (3), we have 2 v final sin θ = v cos ϕ (4) 2 v final cos θ = v (2 sin ϕ ) (5) Squaring these equations and adding them together gives 4 v 2 final = v 2 cos 2 ϕ + v 2 (2 sin ϕ ) 2 (6) = v 2 (sin 2 ϕ 4 sin ϕ + 4 + cos 2 ϕ ) (7) = v 2 (5 4 sin ϕ ) (8) v final = 1 2 v p 5 4 sin ϕ (9) Part B From Equation 4, θ = arcsin v cos ϕ 2 v final (10) = arcsin cos ϕ 5 4 sin ϕ (11) Question 6 Suppose that these objects collide at an angle θ perfectly bisected by the x -axis and travel horizontally post-collision. The angle between each object is ϕ = θ/ 2. Since momentum is conserved, p i = p f (12) 2 mv cos ϕ = 2 m v 3 (13) θ = 2 ϕ = 2 arccos 1 3 (14) = 141 (15) 1
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