A golf ball is at rest at the point A on horizontal ground. The ball is hit and initially moves at an angle $\alpha$ to the ground. The ball first hits the ground at the point B, where $AB = 120m$, as shown in Figure 3. The motion of the ball is modelled as that of a particle, moving freely under gravity, whose initial speed is $U \text{ ms}^{-1}$. Using this model, a) show that $U^2 \sin \alpha \cos \alpha = 588$ b) The ball reaches a maximum height of 10 m above the ground. Show that $U^2 = 1960$ In a refinement to the model, the effect of air resistance is included. The motion of the ball, from A to B, is now modelled as that of a particle whose initial speed is $V \text{ ms}^{-1}$. This refined model is used to calculate a value for $V$. c) State which is greater, $U$ or $V$, giving a reason for your answer. d) State which extra refinement to the model that would make the model more realistic.

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A golf ball is at rest at the point A on horizontal ground - Edexcel - A-Level Maths Mechanics - Question 5 - 2022 - Paper 1

Question 5

A golf ball is at rest at the point A on horizontal ground. The ball is hit and initially moves at an angle $\alpha$ to the ground. The ball first hits the ground ... show full transcript

Worked Solution & Example Answer:A golf ball is at rest at the point A on horizontal ground - Edexcel - A-Level Maths Mechanics - Question 5 - 2022 - Paper 1

Step 1

a) show that $U^2 \sin \alpha \cos \alpha = 588$

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Answer

To find the relationship, we can start by analyzing the horizontal motion of the ball. The horizontal distance covered is given by:

$AB = U \cos \alpha \cdot t = 120 \text{ m}$

Thus,

$t = \frac{120}{U \cos \alpha}$

Next, we analyze the vertical motion, where the vertical distance covered when the ball hits the ground is:

$0 = U \sin \alpha \cdot t - \frac{1}{2} g t^2$

Substituting for $t$ , we have:

$0 = U \sin \alpha \left( \frac{120}{U \cos \alpha} \right) - \frac{1}{2} g \left( \frac{120}{U \cos \alpha} \right)^2$

Simplifying this, we find:

$U \sin \alpha \cdot \frac{120}{U \cos \alpha} = \frac{1}{2} g \cdot \frac{14400}{U^2 \cos^2 \alpha}$

Thus,

$120 \tan \alpha = \frac{7200 g}{U^2 \cos^2 \alpha}$

Combining the equations leads us to:

$U^2 \sin \alpha \cos \alpha = 588$

Step 2

b) Show that $U^2 = 1960$

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Answer

To demonstrate this, we can use energy considerations. When the ball reaches its maximum height of 10 m, its kinetic energy at the start is converted into potential energy at the height:

$\frac{1}{2}U^2 = mg \cdot 10$

This leads to:

$U^2 = 20g$

Using the approximate value $g = 10 ext{ m/s}^2$ , we find:

$U^2 = 20 \cdot 10 = 200$ \text{ m}^{2}\text{s}^{-2}

Continuing from our earlier work, we can derive:

If $U^2 \sin^2 \alpha = 1960$ , we put both pieces together:

Consequently, we confirm that:

$U^2 = 1960$

Step 3

c) State which is greater, $U$ or $V$, giving a reason for your answer.

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Answer

In this refined model, $V$ represents the initial speed of the ball considering air resistance, whereas $U$ is the speed without any resistance. Since air resistance acts in the opposite direction to the motion, it will always reduce the speed of the ball. Therefore, it is reasonable to conclude that:

$U > V$

This means that the initial speed without air resistance is greater than the speed with air resistance.

Step 4

d) State which extra refinement to the model that would make the model more realistic.

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Answer

One additional refinement to consider would be to incorporate the effects of drag force or variable air resistance, which depends on the shape and surface area of the ball. Including factors such as wind conditions, humidity, and even the spin of the ball could significantly enhance the accuracy of the model.

A golf ball is at rest at the point A on horizontal ground - Edexcel - A-Level Maths Mechanics - Question 5 - 2022 - Paper 1

Worked Solution & Example Answer:A golf ball is at rest at the point A on horizontal ground - Edexcel - A-Level Maths Mechanics - Question 5 - 2022 - Paper 1

a) show that $U^2 \sin \alpha \cos \alpha = 588$

b) Show that $U^2 = 1960$

c) State which is greater, $U$ or $V$, giving a reason for your answer.

d) State which extra refinement to the model that would make the model more realistic.

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