CAIE M1 (Mechanics 1) 2018 November

1 A particle of mass 0.2 kg moving in a straight line experiences a constant resistance force of 1.5 N . When the particle is moving at speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), a constant force of magnitude \(F \mathrm {~N}\) is applied to it in the direction in which it is moving. Given that the speed of the particle 5 seconds later is \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the value of \(F\).

2 A high-speed train of mass 490000 kg is moving along a straight horizontal track at a constant speed of \(85 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The engines are supplying 4080 kW of power.

1. Show that the resistance force is 48000 N .
2. The train comes to a hill inclined at an angle \(\theta ^ { \circ }\) above the horizontal, where \(\sin \theta ^ { \circ } = \frac { 1 } { 200 }\). Given that the resistance force is unchanged, find the power required for the train to keep moving at the same constant speed of \(85 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

3 A van of mass 2500 kg descends a hill of length 0.4 km inclined at \(4 ^ { \circ }\) to the horizontal. There is a constant resistance to motion of 600 N and the speed of the van increases from \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as it descends the hill. Find the work done by the van's engine as it descends the hill.

4 Two particles \(A\) and \(B\), of masses \(m \mathrm {~kg}\) and 0.3 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley and the particles hang freely below it. The system is released from rest, with both particles 0.8 m above horizontal ground. Particle \(A\) reaches the ground with a speed of \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the tension in the string during the motion before \(A\) reaches the ground.
2. Find the value of \(m\).

5
\includegraphics[max width=\textwidth, alt={}, center]{98a5537b-d503-4a42-bbfe-0bd221084ee0-06_449_654_260_742} Coplanar forces, of magnitudes \(15 \mathrm {~N} , 25 \mathrm {~N}\) and 30 N , act at a point \(B\) on the line \(A B C\) in the directions shown in the diagram.

1. Find the magnitude and direction of the resultant force.
2. The force of magnitude 15 N is now replaced by a force of magnitude \(F \mathrm {~N}\) acting in the same direction. The new resultant force has zero component in the direction \(B C\). Find the value of \(F\), and find also the magnitude and direction of the new resultant force.

6 A particle is projected from a point \(P\) with initial speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a line of greatest slope \(P Q R\) of a rough inclined plane. The distances \(P Q\) and \(Q R\) are both equal to 0.8 m . The particle takes 0.6 s to travel from \(P\) to \(Q\) and 1 s to travel from \(Q\) to \(R\).

1. Show that the deceleration of the particle is \(\frac { 2 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and hence find \(u\), giving your answer as an exact fraction.
2. Given that the plane is inclined at \(3 ^ { \circ }\) to the horizontal, find the value of the coefficient of friction between the particle and the plane.

7 A particle moves in a straight line starting from rest from a point \(O\). The acceleration of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where $$a = 5.4 - 1.62 t$$

1. Find the positive value of \(t\) at which the velocity of the particle is zero, giving your answer as an exact fraction.
2. Find the velocity of the particle at \(t = 10\) and sketch the velocity-time graph for the first ten seconds of the motion.
3. Find the total distance travelled during the first ten seconds of the motion.
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