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 ms⁻¹. Show that $$v = \\sqrt{2gh}$$ 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 Constant Acceleration - 1D: 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 partic... 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 $v = \sqrt{2gh}$

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Answer

To show that the velocity of the particle as it reaches the ground is given by the formula $v = \sqrt{2gh}$ , we can start from the equations of motion under constant acceleration.

Identify the initial conditions: Since the particle is released from rest, the initial velocity (u) is 0. Therefore, we can use the equation: $v^2 = u^2 + 2as$ Here,
- $u = 0$ (initial velocity)
- $a = g$ (acceleration due to gravity, approximately 9.8 m/s²)
- $s = h$ (the height from which it is dropped)
Substituting values: With these values, the equation becomes: $v^2 = 0^2 + 2gh$
Simplifying: This simplifies to: $v^2 = 2gh$
Taking the square root: Finally, we take the square root of both sides to find v: $v = \sqrt{2gh}$

Step 2

Determine if the teacher's claim is correct.

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Answer

To determine whether the teacher's claim about the machine measuring the velocity is correct, we first calculate the velocity using the given height (h = 18 m) and the formula derived earlier:

Calculate the expected velocity: Using the formula: $v = \sqrt{2gh}$ Substitute $g = 9.8 \, \text{m/s}^2$ and $h = 18 \, \text{m}$ : $v = \sqrt{2 \times 9.8 \times 18}$
- Calculate: $v = \sqrt{352.8}$ $v \approx 18.8 \, \text{m/s}$
Compare measured and calculated values: The student measured $v$ as 20 m/s. When we compare:
- Calculated velocity: 18.8 m/s
- Measured velocity: 20 m/s
Analysis: The student's measured velocity of 20 m/s is greater than the calculated velocity of 18.8 m/s. Therefore, the measurement is inconsistent with the theoretical prediction. This inconsistency suggests a fault in the measurement.
Conclusion: As the calculated value of velocity (18.8 m/s) is significantly less than the measured value (20 m/s), we conclude that the teacher's claim about the machine being faulty is justified.

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 $v = \sqrt{2gh}$

Determine if the teacher's claim is correct.

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