CAIE M1 (Mechanics 1) 2005 November

1 A car travels in a straight line with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). It passes the points \(A , B\) and \(C\), in this order, with speeds \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 } , 7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The distances \(A B\) and \(B C\) are \(d _ { 1 } \mathrm {~m}\) and \(d _ { 2 } \mathrm {~m}\) respectively.

1. Write down an equation connecting
   (a) \(d _ { 1 }\) and \(a\),
   (b) \(d _ { 2 }\) and \(a\).
2. Hence find \(d _ { 1 }\) in terms of \(d _ { 2 }\).

2 A crate of mass 50 kg is dragged along a horizontal floor by a constant force of magnitude 400 N acting at an angle \(\alpha ^ { \circ }\) upwards from the horizontal. The total resistance to motion of the crate has constant magnitude 250 N . The crate starts from rest at the point \(O\) and passes the point \(P\) with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The distance \(O P\) is 20 m . For the crate's motion from \(O\) to \(P\), find

1. the increase in kinetic energy of the crate,
2. the work done against the resistance to the motion of the crate,
3. the value of \(\alpha\).

3
\includegraphics[max width=\textwidth, alt={}, center]{2026cad4-8494-4139-ad21-d8a17ac2b955-2_479_771_1356_687} Each of three light strings has a particle attached to one of its ends. The other ends of the strings are tied together at a point \(A\). The strings are in equilibrium with two of them passing over fixed smooth horizontal pegs, and with the particles hanging freely. The weights of the particles, and the angles between the sloping parts of the strings and the vertical, are as shown in the diagram. Find the values of \(W _ { 1 }\) and \(W _ { 2 }\).
[0pt] [6]

4
\includegraphics[max width=\textwidth, alt={}, center]{2026cad4-8494-4139-ad21-d8a17ac2b955-3_276_570_264_790} A stone slab of mass 320 kg rests in equilibrium on rough horizontal ground. A force of magnitude \(X \mathrm {~N}\) acts upwards on the slab at an angle of \(\theta\) to the vertical, where \(\tan \theta = \frac { 7 } { 24 }\) (see diagram).

1. Find, in terms of \(X\), the normal component of the force exerted on the slab by the ground.
2. Given that the coefficient of friction between the slab and the ground is \(\frac { 3 } { 8 }\), find the value of \(X\) for which the slab is about to slip.

5
\includegraphics[max width=\textwidth, alt={}, center]{2026cad4-8494-4139-ad21-d8a17ac2b955-3_917_1451_1059_347} The diagram shows the displacement-time graph for a car's journey. The graph consists of two curved parts \(A B\) and \(C D\), and a straight line \(B C\). The line \(B C\) is a tangent to the curve \(A B\) at \(B\) and a tangent to the curve \(C D\) at \(C\). The gradient of the curves at \(t = 0\) and \(t = 600\) is zero, and the acceleration of the car is constant for \(0 < t < 80\) and for \(560 < t < 600\). The displacement of the car is 400 m when \(t = 80\).

1. Sketch the velocity-time graph for the journey.
2. Find the velocity at \(t = 80\).
3. Find the total distance for the journey.
4. Find the acceleration of the car for \(0 < t < 80\).

6 A particle \(P\) starts from rest at \(O\) and travels in a straight line. Its velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) is given by \(v = 8 t - 2 t ^ { 2 }\) for \(0 \leqslant t \leqslant 3\), and \(v = \frac { 54 } { t ^ { 2 } }\) for \(t > 3\). Find

1. the distance travelled by \(P\) in the first 3 seconds,
2. an expression in terms of \(t\) for the displacement of \(P\) from \(O\), valid for \(t > 3\),
3. the value of \(v\) when the displacement of \(P\) from \(O\) is 27 m .

7
\includegraphics[max width=\textwidth, alt={}, center]{2026cad4-8494-4139-ad21-d8a17ac2b955-4_601_515_699_815} Two particles \(A\) and \(B\), of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. Particle \(B\) is held on the horizontal floor and particle \(A\) hangs in equilibrium. Particle \(B\) is released and each particle starts to move vertically with constant acceleration of magnitude \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the value of \(a\). Particle \(A\) hits the floor 1.2 s after it starts to move, and does not rebound upwards.
2. Show that \(A\) hits the floor with a speed of \(2.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the gain in gravitational potential energy by \(B\), from leaving the floor until reaching its greatest height. \footnotetext{Every reasonable effort has been made to trace all copyright holders where the publishers (i.e. UCLES) are aware that third-party material has been reproduced.
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