OCR M1 (Mechanics 1) 2009 June

\includegraphics{figure_1} Two perpendicular forces have magnitudes \(x\) N and \(3x\) N (see diagram). Their resultant has magnitude \(6\) N.

1. Calculate \(x\). [3]
2. Find the angle the resultant makes with the smaller force. [3]

The driver of a car accelerating uniformly from rest sees an obstruction. She brakes immediately bringing the car to rest with constant deceleration at a distance of \(6\) m from its starting point. The car travels in a straight line and is in motion for \(3\) seconds.

1. Sketch the \((t, v)\) graph for the car's motion. [2]
2. Calculate the maximum speed of the car during its motion. [3]
3. Hence, given that the acceleration of the car is \(2.4\) m s\(^{-2}\), calculate its deceleration. [4]

\includegraphics{figure_3} The diagram shows a small block \(B\), of mass \(3\) kg, and a particle \(P\), of mass \(0.8\) kg, which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley. \(B\) is held at rest on a horizontal surface, and \(P\) lies on a smooth plane inclined at \(30°\) to the horizontal. When \(B\) is released from rest it accelerates at \(0.2\) m s\(^{-2}\) towards the pulley.

1. By considering the motion of \(P\), show that the tension in the string is \(3.76\) N. [4]
2. Calculate the coefficient of friction between \(B\) and the horizontal surface. [5]

An object is projected vertically upwards with speed \(7\) m s\(^{-1}\). Calculate

1. the speed of the object when it is \(2.1\) m above the point of projection, [3]
2. the greatest height above the point of projection reached by the object, [3]
3. the time after projection when the object is travelling downwards with speed \(5.7\) m s\(^{-1}\). [3]

1. \includegraphics{figure_5_1} A particle \(P\) of mass \(0.5\) kg is projected with speed \(6\) m s\(^{-1}\) on a smooth horizontal surface towards a stationary particle \(Q\) of mass \(m\) kg (see Fig. 1). After the particles collide, \(P\) has speed \(v\) m s\(^{-1}\) in the original direction of motion, and \(Q\) has speed \(1\) m s\(^{-1}\) more than \(P\). Show that \(v(m + 0.5) = -m + 3\). [3]
2. \includegraphics{figure_5_2} \(Q\) and \(P\) are now projected towards each other with speeds \(4\) m s\(^{-1}\) and \(2\) m s\(^{-1}\) respectively (see Fig. 2). Immediately after the collision the speed of \(Q\) is \(v\) m s\(^{-1}\) with its direction of motion unchanged and \(P\) has speed \(1\) m s\(^{-1}\) more than \(Q\). Find another relationship between \(m\) and \(v\) in the form \(v(m + 0.5) = am + b\), where \(a\) and \(b\) are constants. [4]
3. By solving these two simultaneous equations show that \(m = 0.9\), and hence find \(v\). [4]

A block \(B\) of weight \(10\) N is projected down a line of greatest slope of a plane inclined at an angle of \(20°\) to the horizontal. \(B\) travels down the plane at constant speed.

1. 1. Find the components perpendicular and parallel to the plane of the contact force between \(B\) and the plane. [2]
   2. Hence show that the coefficient of friction is \(0.364\), correct to \(3\) significant figures. [2]
2. \includegraphics{figure_6} \(B\) is in limiting equilibrium when acted on by a force of \(T\) N directed towards the plane at an angle of \(45°\) to a line of greatest slope (see diagram). Given that the frictional force on \(B\) acts down the plane, find \(T\). [7]

\includegraphics{figure_7} A sprinter \(S\) starts from rest at time \(t = 0\), where \(t\) is in seconds, and runs in a straight line. For \(0 \leq t \leq 3\), \(S\) has velocity \((6t - t^2)\) m s\(^{-1}\). For \(3 < t \leq 22\), \(S\) runs at a constant speed of \(9\) m s\(^{-1}\). For \(t > 22\), \(S\) decelerates at \(0.6\) m s\(^{-2}\) (see diagram).

1. Express the acceleration of \(S\) during the first \(3\) seconds in terms of \(t\). [2]
2. Show that \(S\) runs \(18\) m in the first \(3\) seconds of motion. [5]
3. Calculate the time \(S\) takes to run \(100\) m. [3]
4. Calculate the time \(S\) takes to run \(200\) m. [7]