Question

We first look at a particle that moves in a one-dimensional potential with form:
? (?) = ?₀ (1/2 ((? / ?) ^ 4) - ((? / ?) ^ 2)),
where ?₀ is a constant with unit Joule and ? a constant with unit meter. We can also imagine
a small sphere influenced by the gravitational acceleration ? that rolls along a roller coaster, where
the height above the ground can be described as:
ℎ (?) = ℎ₀ (1/2 ((? / ?) ^ 4) - ((? / ?) ^ 2)),
so that the potential energy of a mass of mass ? becomes:
? (?) = ??ℎ₀ (1/2 ((? / ?) ^ 4) - ((? / ?) ^ 2))

b) Show that the force acting on a mass of mass ? in position ? is:
? (?) = 2?? (ℎ₀ / ?) * (? / ?− ((? / ?) ^ 3))

c) Find the equilibrium points and characterize them as stable or unstable.

d) You release a particle without initial velocity at a position ?> 0. Where to escape from
that the particle should reach a position with ? <0?

We are now looking at a situation where a particle A with mass ?? = ? is released without initial velocity at
position ?? = −2?. A particle B with mass ?? = 2? is at rest in position ? = −?. The two particles
collide in an inelastic collision with the recovery coefficient
??? = - ((?_A,₁ - ?_B,₁) / (?_A,0 - v_B,0)) = 0.5
where ?_A,0 and ?_A,1 are the velocities of particle A immediately before and after the collision, and correspondingly
for particle B.

e) Find the velocity of particle A just before it collides with particle B at position ? = −?.

f) Find the velocities ?_A,1 and v_B,1 immediately after the collision.

Answer 1