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Figure 3 is a graph of the trajectory of a golf ball after the ball has been hit until it first hits the ground - Edexcel - A-Level Maths Pure - Question 14 - 2021 - Paper 1

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Figure 3 is a graph of the trajectory of a golf ball after the ball has been hit until it first hits the ground. The vertical height, H metres, of the ball above th... show full transcript

Worked Solution & Example Answer:Figure 3 is a graph of the trajectory of a golf ball after the ball has been hit until it first hits the ground - Edexcel - A-Level Maths Pure - Question 14 - 2021 - Paper 1

Step 1

find H in terms of x

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Answer

To determine H in terms of x, we start by knowing that H can be expressed as a quadratic equation:

H=ax2+bx+cH = ax^2 + bx + c
Since we know that H = 3 when x = 0, we can set up our equation to find c:

c=3c = 3
Next, we also have that H = 27 when x = 120, which gives us another equation:

27=a(120)2+b(120)+327 = a(120)^2 + b(120) + 3
This simplifies to:

27=14400a+120b+327 = 14400a + 120b + 3
14400a+120b=24 ag114400a + 120b = 24\ ag{1}

Moreover, we know that when x = 90, H reaches its maximum:

Taking the first derivative, dHdx=2ax+b=0\frac{dH}{dx} = 2ax + b = 0 Therefore, substituting x = 90, we find:

2a(90)+b=0 ag22a(90) + b = 0\ ag{2}

Now solving the equations (1) and (2) will give us the values of a and b.

Step 2

the maximum vertical height of the ball above the ground

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Answer

Using the value we found for a and substituting it back, we can find the maximum height. The maximum height occurs at x = 90:

Hmax=a(90)2+b(90)+3H_{max} = a(90)^2 + b(90) + 3
Substituting the values of a and b into this equation will give the final maximum height.

Step 3

the horizontal distance travelled by the ball, from when it was hit to when it first hits the ground

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Answer

To find the horizontal distance when the ball first hits the ground, we need to set H = 0:

0=ax2+bx+30 = ax^2 + bx + 3
You can solve this quadratic equation using the quadratic formula:** x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
After substituting the coefficients, you can derive the correct horizontal distance directly.

Step 4

other limitation of the model

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Answer

One other limitation of the model is that it assumes that the ball travels in a vacuum, neglecting the effects of air resistance and wind conditions on its trajectory.

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