Two particles P and Q have mass 0.4 kg and 0.6 kg respectively. The particles are initially at rest on a smooth horizontal table. Particle P is given an impulse of magnitude 3 N s in the direction PQ. (a) Find the speed of P immediately before it collides with Q. Immediately after the collision between P and Q, the speed of Q is 5 m s⁻¹. (b) Show that immediately after the collision P is at rest.

A-Level Edexcel Maths Mechanics Moments: Two particles P and Q have mass

Two particles P and Q have mass 0.4 kg and 0.6 kg respectively - Edexcel - A-Level Maths Mechanics - Question 1 - 2008 - Paper 1

Question 1

Two particles P and Q have mass 0.4 kg and 0.6 kg respectively. The particles are initially at rest on a smooth horizontal table. Particle P is given an impulse of m... show full transcript

Worked Solution & Example Answer:Two particles P and Q have mass 0.4 kg and 0.6 kg respectively - Edexcel - A-Level Maths Mechanics - Question 1 - 2008 - Paper 1

Step 1

Find the speed of P immediately before it collides with Q.

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Answer

To find the speed of particle P immediately before it collides with Q, we use the impulse-momentum theorem, which states that the impulse applied to an object is equal to the change in momentum of that object.

Given:

Impulse, $I = 3$ N s
Mass of P, $m_P = 0.4$ kg
Initial velocity of P, $u_P = 0$ (initially at rest)

Using the impulse-momentum formula: $I = m_P(v_P - u_P)$ Substituting the known values:

$3 = 0.4(v_P - 0)$

Solving for $v_P$ gives: $v_P = \frac{3}{0.4} = 7.5\text{ m s}^{-1}$

Thus, the speed of P immediately before it collides with Q is 7.5 m s⁻¹.

Step 2

Show that immediately after the collision P is at rest.

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Answer

After the collision, we can use the principle of conservation of momentum:

The total momentum before the collision must equal the total momentum after the collision.

Let:

Speed of P after the collision be $v_P$
Speed of Q after the collision be $v_Q = 5$ m s⁻¹

Before the collision, the momentum of the system is: $\text{Total momentum before} = m_P v_P + m_Q v_Q\text{ (initially only P is moving)} = 0.4 \cdot 7.5 + 0.6 \cdot 0 = 3\text{ kg m s}^{-1}$

After the collision: $\text{Total momentum after} = m_P v_P + m_Q v_Q = 0.4 v_P + 0.6 \cdot 5$

Setting the two equal: $3 = 0.4 v_P + 3$

Rearranging the equation gives: $0.4 v_P = 3 - 3\rightarrow 0.4 v_P = 0\rightarrow v_P = 0$

Thus, the speed of P immediately after the collision is 0 m s⁻¹, which confirms that P is at rest.

Two particles P and Q have mass 0.4 kg and 0.6 kg respectively - Edexcel - A-Level Maths Mechanics - Question 1 - 2008 - Paper 1

Worked Solution & Example Answer:Two particles P and Q have mass 0.4 kg and 0.6 kg respectively - Edexcel - A-Level Maths Mechanics - Question 1 - 2008 - Paper 1

Find the speed of P immediately before it collides with Q.

Show that immediately after the collision P is at rest.

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