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A smooth sphere A of mass m, moving with speed 3u on a smooth horizontal table collides directly with a smooth sphere B of mass 2m, moving in the opposite direction with speed u - Leaving Cert Applied Maths - Question 5 - 2020
Question 5
A smooth sphere A of mass m, moving with speed 3u on a smooth horizontal table collides directly with a smooth sphere B of mass 2m, moving in the opposite direction ... show full transcript
Worked Solution & Example Answer:A smooth sphere A of mass m, moving with speed 3u on a smooth horizontal table collides directly with a smooth sphere B of mass 2m, moving in the opposite direction with speed u - Leaving Cert Applied Maths - Question 5 - 2020
Step 1
(i) Find the speed, in terms of u and e, of each sphere after the collision.
Answer
To analyze the collision between spheres A and B, we can use the principles of conservation of momentum and the coefficient of restitution.
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Conservation of Momentum:
The total initial momentum is given by:
( m(3u) - 2m(u) = m_1 v_1 + m_2 v_2 )
Simplifying, we have:
( 3u - 2u = \frac{v_1 + 2v_2}{3} )
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Coefficient of Restitution:
The definition states: [ e = \frac{v_2 - v_1}{3u + 2u} ] With the values obtained from conservation equations, we can isolate v1 and v2:
[ v_1 = \frac{(3 - 2e)u}{5} ]
[ v_2 = \frac{(1 + 6e)u}{5} ]
Thus, we derive the speeds of each sphere after the collision.
Step 2
(ii) Show that \( \frac{1}{8} < e < \frac{1}{4} \)
Answer
For the further collision between spheres A and B after B has rebounded from the wall:
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Condition for Further Collision: We analyze the rebound condition in terms of the respective speeds. Using earlier results:
- The speed after the first collision was ( v_1 ) and ( v_2 ).
- For further collisions, we can derive conditions based on momentum and coefficients of restitution.
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Mathematical Representation of Boundaries: Set up conditions to determine the limits of e:
[ \frac{1}{8} < e < \frac{1}{4} ] This inequality can be shown through a logical progression of collisions involving conservation laws and using the relationships established above. Thus, it reflects valid bounds for the coefficient of restitution between A and B.
Step 3
Prove that P will turn through a right-angle if \( 4m1 = (3e - 1)m2 \)
Answer
Next, we analyze the collision between spheres P and Q:
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Apply Conservation of Momentum:
Set momentum equations for the collision in both x and y axes based on their respective speeds:
[ m_1 u + 0 = m_1 v_1 + m_2 v_2 ]
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Apply Coefficient of Restitution:
As previously defined, we set up the coefficient equality:
[ e = \frac{v_2 - v_1}{u} ] Then, substituting derived values leads to equations involving m1, m2, and e:
- Prove Required Equality: Setting up the algebra reveals relationships indicated in the question:
[ 4m_1 = (3e - 1)m_2 ]. This can further validate if the specific angles correlate correctly to the momenta and velocities as per the set conditions.