Quadratic trajectory/projectile model

A question is this type if and only if a physical trajectory (ball, arrow, stone) is modelled by a quadratic function H(x) or h(t), and the question asks to find maximum height, range, or interpret constants of the model.

3 questions · Moderate -0.8

1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points
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Edexcel Paper 1 Specimen Q11
9 marks Moderate -0.8
11. An archer shoots an arrow. The height, \(H\) metres, of the arrow above the ground is modelled by the formula $$H = 1.8 + 0.4 d - 0.002 d ^ { 2 } , \quad d \geqslant 0$$ where \(d\) is the horizontal distance of the arrow from the archer, measured in metres.
Given that the arrow travels in a vertical plane until it hits the ground,
  1. find the horizontal distance travelled by the arrow, as given by this model.
  2. With reference to the model, interpret the significance of the constant 1.8 in the formula.
  3. Write \(1.8 + 0.4 d - 0.002 d ^ { 2 }\) in the form $$A - B ( d - C ) ^ { 2 }$$ where \(A , B\) and \(C\) are constants to be found. It is decided that the model should be adapted for a different archer.
    The adapted formula for this archer is $$H = 2.1 + 0.4 d - 0.002 d ^ { 2 } , \quad d \geqslant 0$$ Hence or otherwise, find, for the adapted model
    1. the maximum height of the arrow above the ground.
    2. the horizontal distance, from the archer, of the arrow when it is at its maximum height.
Edexcel Paper 2 2018 June Q8
7 marks Moderate -0.8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{580fc9b9-d78c-4a86-91fc-22638cb5186d-20_540_1465_294_301} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 is a graph showing the trajectory of a rugby ball. The height of the ball above the ground, \(H\) metres, has been plotted against the horizontal distance, \(x\) metres, measured from the point where the ball was kicked. The ball travels in a vertical plane. The ball reaches a maximum height of 12 metres and hits the ground at a point 40 metres from where it was kicked.
  1. Find a quadratic equation linking \(H\) with \(x\) that models this situation. The ball passes over the horizontal bar of a set of rugby posts that is perpendicular to the path of the ball. The bar is 3 metres above the ground.
  2. Use your equation to find the greatest horizontal distance of the bar from \(O\).
  3. Give one limitation of the model.
AQA AS Paper 2 2020 June Q11
11 marks Moderate -0.8
A fire crew is tackling a grass fire on horizontal ground. The crew directs a single jet of water which flows continuously from point \(A\). \includegraphics{figure_11} The path of the jet can be modelled by the equation $$y = -0.0125x^2 + 0.5x - 2.55$$ where \(x\) metres is the horizontal distance of the jet from the fire truck at \(O\) and \(y\) metres is the height of the jet above the ground. The coordinates of point \(A\) are \((a, 0)\)
    1. Find the value of \(a\). [3 marks]
    2. Find the horizontal distance from \(A\) to the point where the jet hits the ground. [1 mark]
  2. Calculate the maximum vertical height reached by the jet. [4 marks]
  3. A vertical wall is located 11 metres horizontally from \(A\) in the direction of the jet. The height of the wall is 2.3 metres. Using the model, determine whether the jet passes over the wall, stating any necessary modelling assumption. [3 marks]