A small uniform sphere A, of mass 2m, is moving with speed u on a smooth horizontal surface when it collides directly with a small uniform sphere B, of mass m, which is at rest. The spheres have equal radii and the coefficient of restitution between them is e. Find expressions for the speeds of A and B immediately after the collision. Subsequently B collides with a vertical wall which is perpendicular to the direction of motion of B. The coefficient of restitution between B and the wall is 0.4. After B has collided with the wall, the speeds of A and B are equal. Find e. Initially B is at a distance d from the wall. Find the distance of B from the wall when it next collides with A.

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A small uniform sphere A, of mass 2m, is moving with speed u on a smooth horizontal surface when it collides directly with a small uniform sphere B, of mass m, which is at rest - CIE - A-Level Further Maths - Question 2 - 2015 - Paper 1

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A small uniform sphere A, of mass 2m, is moving with speed u on a smooth horizontal surface when it collides directly with a small uniform sphere B, of mass m, which... show full transcript

Worked Solution & Example Answer:A small uniform sphere A, of mass 2m, is moving with speed u on a smooth horizontal surface when it collides directly with a small uniform sphere B, of mass m, which is at rest - CIE - A-Level Further Maths - Question 2 - 2015 - Paper 1

Step 1

Find expressions for the speeds of A and B immediately after the collision.

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Answer

To find the speeds of A and B after the collision, we apply the conservation of momentum. Before the collision, the total momentum is:

$p_{initial} = 2mu$

After the collision, let the speed of A be $v_A$ and the speed of B be $v_B$ . Using conservation of momentum:

$2mu = 2mv_A + mv_B$

This simplifies to:

$v_A + \frac{1}{2}v_B = u ag{1}$

Next, we use the coefficient of restitution e, which is defined as:

$e = \frac{v_B - v_A}{u}$

From this, we can express $v_B$ in terms of $v_A$ :

$v_B = v_A + eu$

Substituting $v_B$ in equation (1):

$v_A + \frac{1}{2}(v_A + eu) = u$

Simplifying this gives:

$\frac{3}{2}v_A + \frac{1}{2}eu = u$

Solving for $v_A$ yields:

$v_A = \frac{2(1-e)}{3}u$

And substituting back to find $v_B$ :

$v_B = \frac{2}{3}u + eu$

Thus, the expressions are:

$v_A = \frac{2(1-e)}{3}u, \quad v_B = \frac{2(1+e)}{3}u$

Step 2

Find e.

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Answer

After B has collided with the wall, the coefficient of restitution between B and the wall is given as 0.4. Since the speeds of A and B are equal after this collision, we denote this common speed as $v$ :

Setting $v_A = v_B = v$ , we use the coefficient of restitution to relate to the speed of B after colliding with the wall:

$v = -0.4v_B$

Since $v_B = \frac{2(1+e)}{3}u$ prior to hitting the wall, we have:

$v = -0.4 \left( \frac{2(1+e)}{3}u \right)$

Setting these equal gives:

$\frac{2(1-e)}{3}u = -0.4 \left( \frac{2(1+e)}{3}u \right)$

Cancelling common terms and simplifying leads to:

$1-e = -0.4(1+e) \Rightarrow 1-e = -0.4 - 0.4e$

Solving for e:

$1 + 0.4 = e + 0.4e\Rightarrow e(1 + 0.4) = 1.4 \Rightarrow e = \frac{1.4}{1.4} = 0.714$

Step 3

Find the distance of B from the wall when it next collides with A.

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Answer

Let the distance from B to the wall be denoted as $d$ . After colliding with the wall, the speed of B is:

$v_B' = -0.4v_B$

Using the expression for speed before the wall collision:

$v_B = \frac{2(1+0.714)}{3}u = \frac{5.428}{3}u$

Thus, after the collision:

$v_B' = -0.4 \cdot \frac{5.428}{3}u = -0.572u$

Now, to find the distance $x$ traveled by B before colliding with A again, we set:

$x = d - v_B' t$

Assuming uniform motion, substituting the equation will yield:

Using the relationship:

ightarrow d = x + 0.3$$ Thus, the distance of B from the wall when it next collides with A: $$x = d - 0.3$$

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A small uniform sphere A, of mass 2m, is moving with speed u on a smooth horizontal surface when it collides directly with a small uniform sphere B, of mass m, which is at rest - CIE - A-Level Further Maths - Question 2 - 2015 - Paper 1

Worked Solution & Example Answer:A small uniform sphere A, of mass 2m, is moving with speed u on a smooth horizontal surface when it collides directly with a small uniform sphere B, of mass m, which is at rest - CIE - A-Level Further Maths - Question 2 - 2015 - Paper 1

Find expressions for the speeds of A and B immediately after the collision.

Find e.

Find the distance of B from the wall when it next collides with A.

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