Show that the x coordinates of the turning points of the curve with equation y = f(x) satisfy the equation tan x = 4 - Edexcel - A-Level Maths Pure - Question 12 - 2019 - Paper 1

To sketch the graph of H(t), it's important to understand that as t increases, the factor $e^{-0.025t}$ will decrease towards zero, which indicates that the height H(t) will oscillate between values defined by the amplitude modulated by this exponential decay.

1. For small values of t, H(t) will reach its maximum due to the sin function oscillating between -1 and 1.
2. As t continues to increase, the oscillations will diminish in amplitude, leading to a damping effect.

This means the overall shape of the curve will display diminishing loops, showcasing that while the oscillations continue, their heights decrease over time.

To calculate the maximum height of the ball above the ground, we need to evaluate the function H(t) at its peak:

1. From previous calculations, we know that at points where sin t = ±1, the height is maximized:

   $H(t) = |10e^{-0.025t}|$
   At $t = 0$ (the first kick),
   $H(0) = |10e^{0}| = 10 ext{ meters}$
   After the first bounce, to find when the ball reaches the same height again: $H(t) = 10e^{-0.025t}$

At the first bounce, we apply the period of the sine function: The maximum height occurs at approximately $t ext{ when the next peak of sin t is reached.}$ Thus, solving for when sin t = 1, which can be calculated, leads to H(x) diminishing to a height of about 3.18 meters on that first cycle.

This model should not be used to predict the timing of each bounce because it oversimplifies the behavior of a ball's motion in real-world conditions. Actual bounces depend on various factors such as:

1. Air resistance, which affects how high and how long the ball travels after being kicked.
2. Energy loss due to deformation of the ball and the surface on which it lands.
3. The effect of surface material (hard or soft) on the bounce height.

Thus, while the model may work for height calculations, it does not accurately represent the dynamics involved in real-life bounces, making it unreliable for timing predictions.