Two particles A and B are released from rest from different starting points above a horizontal surface. A is released from a height of h metres. B is released at a time t seconds after A from a height of kh metres, where 0 < k < 1. Both particles land on the surface 5 seconds after A was released. Assuming any resistance forces may be ignored, prove that t = 5(1 - √k) Fully justify your answer.

A-Level AQA Maths Pure Proof: Two particles A and B are releas

Two particles A and B are released from rest from different starting points above a horizontal surface - AQA - A-Level Maths Pure - Question 16 - 2020 - Paper 2

Question 16

Two particles A and B are released from rest from different starting points above a horizontal surface. A is released from a height of h metres. B is released at a... show full transcript

Worked Solution & Example Answer:Two particles A and B are released from rest from different starting points above a horizontal surface - AQA - A-Level Maths Pure - Question 16 - 2020 - Paper 2

Step 1

Find h in terms of g

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Answer

For particle A, using the equation of motion:

$s = ut + \frac{1}{2} a t^2$

where $u = 0$ , $a = g$ , and $t = 5$ . Thus,

$h = \frac{1}{2} g t^2$

Substituting the value of $t$ :

$h = \frac{1}{2} g (5^2) = \frac{25g}{2}$

Step 2

Use kh = \frac{1}{2} gt^2 for particle B

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Answer

For particle B, it is released at time $t$ after A:

Using the formula:

$kh = \frac{1}{2} g (5 - t)^2$

Substituting the expression for $h$ from particle A:

$k \left( \frac{25g}{2} \right) = \frac{1}{2} g (5 - t)^2$

This simplifies to:

$25k = (5 - t)^2$

Step 3

Deduce time for B + t = 5

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Answer

Taking the square root of both sides:

$5 - t = 5\sqrt{k}$

Rearranging gives:

$t = 5 - 5\sqrt{k}$

Thus:

$t = 5(1 - \sqrt{k})$

Step 4

Final Result

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Answer

This shows that the time for particle B to reach the surface can be expressed as:

$t = 5(1 - \sqrt{k})$

which completes the proof.

Two particles A and B are released from rest from different starting points above a horizontal surface - AQA - A-Level Maths Pure - Question 16 - 2020 - Paper 2

Worked Solution & Example Answer:Two particles A and B are released from rest from different starting points above a horizontal surface - AQA - A-Level Maths Pure - Question 16 - 2020 - Paper 2

Find h in terms of g

Use kh = \frac{1}{2} gt^2 for particle B

Deduce time for B + t = 5

Final Result

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