Two uniform small smooth spheres A and B, of equal radii, have masses 2m and m respectively. They lie at rest on a smooth horizontal plane. Sphere A is projected directly towards B with speed u. After the collision B goes on to collide directly with a fixed smooth vertical barrier, before colliding with A again. The coefficient of restitution between A and B is $rac{1}{3}$ and the coefficient of restitution between B and the barrier is e. After the second collision between A and B, the speed of B is five times the speed of A. Find the two possible values of e.

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Two uniform small smooth spheres A and B, of equal radii, have masses 2m and m respectively - CIE - A-Level Further Maths - Question 5 - 2013 - Paper 1

Question 5

Two uniform small smooth spheres A and B, of equal radii, have masses 2m and m respectively. They lie at rest on a smooth horizontal plane. Sphere A is projected dir... show full transcript

Worked Solution & Example Answer:Two uniform small smooth spheres A and B, of equal radii, have masses 2m and m respectively - CIE - A-Level Further Maths - Question 5 - 2013 - Paper 1

Step 1

Use conservation of momentum

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Answer

Using the principle of conservation of momentum for the first collision: $2m u + m v_B = 2mv_A + mv_B'$ where $v_B'$ is the speed of B after the collision.

Step 2

Use restitution for the first collision

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Answer

The coefficient of restitution (e) gives us $e = \frac{v_A - v_B'}{u - v_B}$ . Substituting the known values from the problem, we can find the relation between $v_A$ and $v_B$ .

Step 3

Solve for v_A and v_B after the collision

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Answer

From the momentum equation: $2m u = 2m v_A + m v_B'$ gives us $v_A = \frac{2u - v_B'}{2}$ . Let’s plug this into the restitution equation to solve both $v_A$ and $v_B$ .

Step 4

Find speed of B after striking the barrier

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Answer

After striking the vertical barrier, using the restitution coefficient we can find the new speed of B: $v_B'' = e v_B'$ .

Step 5

Use conservation of momentum after collisions

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Answer

For the final collision between A and B: $m v_B'' = m v_A'' + 2m v_A$ , applying momentum conservation allows us to relate the speeds.

Step 6

Substituting known speeds

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Given that the speed of B after the second collision is five times the speed of A: $v_B'' = 5 v_A''$ , we can substitute and rearrange equations to find the values of e.

Step 7

Final calculations for e

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Answer

Solving the final relationships yields two possible values for e: $e = \frac{5}{9} \text{ or } \frac{2}{3}$ . These represent the two possible coefficients of restitution.

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Two uniform small smooth spheres A and B, of equal radii, have masses 2m and m respectively - CIE - A-Level Further Maths - Question 5 - 2013 - Paper 1

Worked Solution & Example Answer:Two uniform small smooth spheres A and B, of equal radii, have masses 2m and m respectively - CIE - A-Level Further Maths - Question 5 - 2013 - Paper 1

Use conservation of momentum

Use restitution for the first collision

Solve for v_A and v_B after the collision

Find speed of B after striking the barrier

Use conservation of momentum after collisions

Substituting known speeds

Final calculations for e

Join the A-Level students using SimpleStudy...

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