Standard +0.8 This is a standard Further Maths FM1 sequential collision problem requiring conservation of momentum and Newton's restitution law applied twice, followed by an inequality condition for direction reversal. While it involves multiple steps and algebraic manipulation with a parameter m, the techniques are routine for FM1 students and follow a predictable structure. The difficulty is elevated above A-level core mechanics (hence positive score) but remains a textbook-style exercise without requiring novel insight.
2
\includegraphics[max width=\textwidth, alt={}, center]{60f72141-4a99-4907-93b1-adb0cd66948e-2_211_1276_1427_365}
Three particles \(A , B\) and \(C\) are free to move in the same straight line on a large smooth horizontal surface. Their masses are \(1.2 \mathrm {~kg} , 1.8 \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively (see diagram). The coefficient of restitution in collisions between any two of them is \(\frac { 3 } { 4 }\). Initially, \(B\) and \(C\) are at rest and \(A\) is moving with a velocity of \(4.0 \mathrm {~ms} ^ { - 1 }\) towards \(B\).
a) Show that immediately after the collision between \(A\) and \(B\) the speed of \(B\) is \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
b) Find the velocity of \(A\) immediately after this collision.
\(B\) subsequently collides with \(C\).
c) Find, in terms of \(m\), the velocity of \(B\) after its collision with \(C\).
d) Given that the direction of motion of \(B\) is reversed by the collision with \(C\), find the range of possible values of \(m\).
The car is attached to a trailer of mass 200 kg by a light rigid horizontal tow bar. The greatest steady speed of the car and trailer on the road is now \(30 \mathrm {~ms} ^ { - 1 }\). The resistance to motion of the trailer may also be assumed constant.
(b) Find the magnitude of the resistance force on the trailer.
The car and trailer again travel along the road. At one instant their speed is \(15 \mathrm {~ms} ^ { - 1 }\) and their acceleration is \(0.57 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
(c) (i) Find the power of the engine of the car at this instant.
(ii) Find the magnitude of the tension in the tow bar at this instant.
In a refined model of the motion of the car and trailer the resistance to the motion of each is assumed to be zero until they reach a speed of \(10 \mathrm {~ms} ^ { - 1 }\). When the speed is \(10 \mathrm {~ms} ^ { - 1 }\) or above the same constant resistance forces as in the first model are assumed to apply to each.
The car and trailer start at rest on the road and accelerate, using maximum power.
(d) Without carrying out any further calculations,
(i) explain whether the time taken to attain a speed of \(20 \mathrm {~m} ^ { - 1 }\) would be predicted to be lower, the same or higher using the refined model compared with the original model,
(ii) explain whether the greatest steady speed of the system would be predicted to be lower, the same or higher using the refined model compared with the original model.
2\\
\includegraphics[max width=\textwidth, alt={}, center]{60f72141-4a99-4907-93b1-adb0cd66948e-2_211_1276_1427_365}
Three particles $A , B$ and $C$ are free to move in the same straight line on a large smooth horizontal surface. Their masses are $1.2 \mathrm {~kg} , 1.8 \mathrm {~kg}$ and $m \mathrm {~kg}$ respectively (see diagram). The coefficient of restitution in collisions between any two of them is $\frac { 3 } { 4 }$. Initially, $B$ and $C$ are at rest and $A$ is moving with a velocity of $4.0 \mathrm {~ms} ^ { - 1 }$ towards $B$.\\
a) Show that immediately after the collision between $A$ and $B$ the speed of $B$ is $2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.\\
b) Find the velocity of $A$ immediately after this collision.\\
$B$ subsequently collides with $C$.\\
c) Find, in terms of $m$, the velocity of $B$ after its collision with $C$.\\
d) Given that the direction of motion of $B$ is reversed by the collision with $C$, find the range of possible values of $m$.
The car is attached to a trailer of mass 200 kg by a light rigid horizontal tow bar. The greatest steady speed of the car and trailer on the road is now $30 \mathrm {~ms} ^ { - 1 }$. The resistance to motion of the trailer may also be assumed constant.\\
(b) Find the magnitude of the resistance force on the trailer.
The car and trailer again travel along the road. At one instant their speed is $15 \mathrm {~ms} ^ { - 1 }$ and their acceleration is $0.57 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.\\
(c) (i) Find the power of the engine of the car at this instant.\\
(ii) Find the magnitude of the tension in the tow bar at this instant.
In a refined model of the motion of the car and trailer the resistance to the motion of each is assumed to be zero until they reach a speed of $10 \mathrm {~ms} ^ { - 1 }$. When the speed is $10 \mathrm {~ms} ^ { - 1 }$ or above the same constant resistance forces as in the first model are assumed to apply to each.
The car and trailer start at rest on the road and accelerate, using maximum power.\\
(d) Without carrying out any further calculations,\\
(i) explain whether the time taken to attain a speed of $20 \mathrm {~m} ^ { - 1 }$ would be predicted to be lower, the same or higher using the refined model compared with the original model,\\
(ii) explain whether the greatest steady speed of the system would be predicted to be lower, the same or higher using the refined model compared with the original model.
\hfill \mbox{\textit{OCR FM1 AS 2021 Q2 [11]}}