In a school experiment, a particle, of mass m kilograms, is released from rest at a point h metres above the ground. At the instant it reaches the ground, the particle has velocity v m s⁻¹. Show that v = √(2gh) A student correctly used h = 18 and measured v as 20. The student’s teacher claims that the machine measuring the velocity must have been faulty. Determine if the teacher’s claim is correct. Fully justify your answer.

A-Level AQA Maths Mechanics Forces: In a school experiment, a partic

In a school experiment, a particle, of mass m kilograms, is released from rest at a point h metres above the ground - AQA - A-Level Maths Mechanics - Question 13 - 2019 - Paper 2

Question 13

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In a school experiment, a particle, of mass m kilograms, is released from rest at a point h metres above the ground. At the instant it reaches the ground, the part... show full transcript

Worked Solution & Example Answer:In a school experiment, a particle, of mass m kilograms, is released from rest at a point h metres above the ground - AQA - A-Level Maths Mechanics - Question 13 - 2019 - Paper 2

Step 1

Show that v = √(2gh)

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Answer

To derive the equation for the velocity of the particle just before it reaches the ground, we can use the equations of motion. Considering that the initial velocity (u) is 0 and the acceleration (a) is due to gravity (g), we have:

From the second equation of motion:
v^2 = u^2 + 2as

Substituting the known values:
v^2 = 0 + 2gh

Thus, we arrive at the equation:
v = √(2gh).

This confirms the relationship between the velocity of the particle just before impact and the height from which it is released.

Step 2

Determine if the teacher’s claim is correct.

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Answer

We have the values of h = 18 m and the measured velocity v = 20 m/s. We can use the derived equation to determine what the theoretical velocity should be:

Using g ≈ 9.8 m/s², we substitute into the equation:

$v = √(2gh) \Rightarrow v = √(2 \times 9.8 \times 18)$

Calculating this, we find:

$v = √(352.8) \approx 18.8 \, m/s$

Now, comparing the two values:

Measured velocity: 20 m/s
Calculated velocity: 18.8 m/s

Since the measured velocity (20 m/s) is greater than the expected theoretical value (18.8 m/s), it indicates a discrepancy. Therefore, the teacher's claim that the machine measuring velocity is faulty is indeed correct as the measured value does not align with the expected value.

In a school experiment, a particle, of mass m kilograms, is released from rest at a point h metres above the ground - AQA - A-Level Maths Mechanics - Question 13 - 2019 - Paper 2

Worked Solution & Example Answer:In a school experiment, a particle, of mass m kilograms, is released from rest at a point h metres above the ground - AQA - A-Level Maths Mechanics - Question 13 - 2019 - Paper 2

Show that v = √(2gh)

Determine if the teacher’s claim is correct.

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