SPS SPS SM Mechanics 2022 February — Question 12 10 marks

Exam BoardSPS
ModuleSPS SM Mechanics (SPS SM Mechanics)
Year2022
SessionFebruary
Marks10
TopicConstant acceleration (SUVAT)
TypeSketch velocity-time graph
DifficultyStandard +0.3 This is a standard kinematics problem involving two particles with constant acceleration. It requires routine application of SUVAT equations, sketching velocity-time graphs, and solving a quadratic equation. The multi-part structure and need to set up equations for when positions are equal adds some complexity, but all techniques are straightforward and commonly practiced. Slightly easier than average due to the guided structure and standard methods required.
Spec3.02b Kinematic graphs: displacement-time and velocity-time3.02d Constant acceleration: SUVAT formulae
Answer all the questions. Two cyclists, \(A\) and \(B\), are cycling along the same straight horizontal track. The cyclists are modelled as particles and the motion of the cyclists is modelled as follows: • At time \(t = 0\), cyclist \(A\) passes through the point \(O\) with speed \(2\text{ms}^{-1}\) • Cyclist \(A\) is moving in a straight line with constant acceleration \(2\text{ms}^{-2}\) • At time \(t = 2\) seconds, cyclist \(B\) starts from rest at \(O\) • Cyclist \(B\) moves with constant acceleration \(6\text{ms}^{-2}\) along the same straight line and in the same direction as cyclist \(A\) • At time \(t = T\) seconds, \(B\) overtakes \(A\) at the point \(X\) Using the model,
  1. sketch, on the same axes, for the interval from \(t = 0\) to \(t = T\) seconds, • a velocity-time graph for the motion of \(A\) • a velocity-time graph for the motion of \(B\) [2]
  2. explain why the two graphs must cross before time \(t = T\) seconds, [1]
  3. find the time when \(A\) and \(B\) are moving at the same speed, [2]
  4. find the distance \(OX\) [5]
Answer all the questions.

Two cyclists, $A$ and $B$, are cycling along the same straight horizontal track.

The cyclists are modelled as particles and the motion of the cyclists is modelled as follows:

• At time $t = 0$, cyclist $A$ passes through the point $O$ with speed $2\text{ms}^{-1}$

• Cyclist $A$ is moving in a straight line with constant acceleration $2\text{ms}^{-2}$

• At time $t = 2$ seconds, cyclist $B$ starts from rest at $O$

• Cyclist $B$ moves with constant acceleration $6\text{ms}^{-2}$ along the same straight line and in the same direction as cyclist $A$

• At time $t = T$ seconds, $B$ overtakes $A$ at the point $X$

Using the model,

\begin{enumerate}[label=(\alph*)]
\item sketch, on the same axes, for the interval from $t = 0$ to $t = T$ seconds,
• a velocity-time graph for the motion of $A$
• a velocity-time graph for the motion of $B$
[2]

\item explain why the two graphs must cross before time $t = T$ seconds,
[1]

\item find the time when $A$ and $B$ are moving at the same speed,
[2]

\item find the distance $OX$
[5]
\end{enumerate}

\hfill \mbox{\textit{SPS SPS SM Mechanics 2022 Q12 [10]}}
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