H1. A particle 1 of mass m₁ is attached to one end of a spring with constant k at position x = 0. It is initially at rest, with velocity u₁ = 0. The other end of the spring is attached to a fixed wall at position x = - where is the length of the spring at rest. The whole setup is horizontal along the x-axis. Another particle, 2, of mass m₂ travels towards particle 1 with constant velocity u₂ (note that u₂ <0 in this setup). The two particles undergo a collision with coefficient of restitution 0
H1. A particle 1 of mass m₁ is attached to one end of a spring with constant k at position x = 0. It is initially at rest, with velocity u₁ = 0. The other end of the spring is attached to a fixed wall at position x = - where is the length of the spring at rest. The whole setup is horizontal along the x-axis. Another particle, 2, of mass m₂ travels towards particle 1 with constant velocity u₂ (note that u₂ <0 in this setup). The two particles undergo a collision with coefficient of restitution 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:H1. A particle 1 of mass m₁ is attached to one end of a spring with constant k at position
x = 0. It is initially at rest, with velocity u₁ = 0. The other end of the spring is attached to
a fixed wall at position x = - where is the length of the spring at rest. The whole setup
is horizontal along the x-axis. Another particle, 2, of mass m₂ travels towards particle 1 with
constant velocity u₂ (note that u₂ <0 in this setup). The two particles undergo a collision with
coefficient of restitution 0 <e < 1.
(a) Using the one-dimensional equations of collision presented in the lectures show that imme-
diately after collision the velocity of particle 1 is given by
m242
V₁ = (1+e);
m₁ + m₂
Find the velocity v2 of particle 2 immediately after collision. What happens to particle 2
m₁ = m₂ and the collision is elastic (e = 1)?
(b) Immediately after collision, particle 1 has velocity v₁ and is at position x = 0, which we
take as initial conditions for its subsequent motion under the influence of the spring (we assume
that there is no friction or resistance). Use energy arguments to determine how close particle
1 comes to the wall. In other words, find the minimum value of x(t) in terms of e, k, u₂, m₁
and m₂. If you aim to send particle 1 as close as possible to the wall by acting only on the
coefficient of restitution e, what value of e would you pick?
(We implicitly assume that the length of the spring is sufficiently long to ensure that the
previous answer makes sense, i.e., is large enough so that particle 1 does not crash into the
wall)
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