| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Topic | Geometric Sequences and Series |
| Type | Compound growth applications |
| Difficulty | Moderate -0.3 This is a straightforward geometric progression application with clear structure: part (a)(i) is simple verification requiring one calculation (3 × 0.6² = 1.08), part (a)(ii) involves summing a finite number of terms with explicit counting, part (b) applies the standard infinite GP sum formula, and part (c) requires basic physical reasoning about energy loss. While it requires multiple steps and understanding of series, all techniques are standard A-level material with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<11.04k Modelling with sequences: compound interest, growth/decay |
A ball is released from rest from a height of 5 m and bounces repeatedly on horizontal ground.
After hitting the ground for the first time, the ball rises to a maximum height of 3 m.
In a model for the motion of the ball
• the maximum height after each bounce is 60% of the previous maximum height
• the motion takes place in a vertical line
(a) Using the model
\begin{enumerate}[label=(\roman*)]
\item show that the maximum height after the 3rd bounce is 1.08 m,
\item find the total distance the ball travels from release to when the ball hits the ground for the 5th time.
\end{enumerate} [3]
According to the model, after the ball is released, there is a limit, $D$ metres, to the total distance the ball will travel.
(b) Find the value of $D$ [2]
With reference to the model,
(c) give a reason why, in reality, the ball will not travel $D$ metres in total. [1]
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q7 [6]}}