A rugby player runs with the ball directly toward his opponent's goal, along the positive direction of an x axis. He can legally
pass the ball to a teammate as long as the ball's velocity relative to the field does not have a positive x component. Suppose the player runs at speed 3.5 m/s relative to the field while he passes the ball with velocity
v⃗ BP
relative to himself. If
v⃗ BP
has magnitude 6.0 m/s, what is the smallest angle it can have for the pass to be legal?
v⃗ BP
relative to himself. If
v⃗ BP
has magnitude 6.0 m/s, what is the smallest angle it can have for the pass to be legal?
1 answer:
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A)
The total energy of the system is equal to the maximum elastic potential energy, that is achieved when the displacement is equal to the amplitude (x=A):
(1)
where k is the spring constant.
The total energy, which is conserved, at any other point of the motion is the sum of elastic potential energy and kinetic energy:
(2)
where x is the displacement, m the mass, and v the speed.
We want to know the displacement x at which the elastic potential energy is 1/3 of the kinetic energy:
Using (2) we can rewrite this as
And using (1), we find
Substituting into the last equation, we find the value of x:
B)
In this case, the kinetic energy is 1/10 of the total energy:
Since we have
we can write
And so we find: