Motion down smooth slope

7 Two particles \(P\) and \(Q\) move on a line of greatest slope of a smooth inclined plane. The particles start at the same instant and from the same point, each with speed \(1.3 \mathrm {~ms} ^ { - 1 }\). Initially \(P\) moves down the plane and \(Q\) moves up the plane. The distance between the particles \(t\) seconds after they start to move is \(d \mathrm {~m}\).

1. Show that \(d = 2.6 t\). When \(t = 2.5\) the difference in the vertical height of the particles is 1.6 m . Find
2. the acceleration of the particles down the plane,
3. the distance travelled by \(P\) when \(Q\) is at its highest point.

1 A particle \(P\) is released from rest at a point on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. Find the speed of \(P\)

1. when it has travelled 0.9 m ,
2. 0.8 s after it is released.

2 A particle of mass 0.5 kg starts from rest and slides down a line of greatest slope of a smooth plane. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal.

1. Find the time taken for the particle to reach a speed of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the particle has travelled 3 m down the slope from its starting point, it reaches rough horizontal ground at the bottom of the slope. The frictional force acting on the particle is 1 N .
2. Find the distance that the particle travels along the ground before it comes to rest.

5 A particle is projected from a point \(O\) on a plane which is inclined at an angle \(\theta\) to the horizontal. The particle is projected down the plane with velocity \(u\) at an angle \(\alpha\) above the plane. The particle first strikes the plane at a point \(P\), as shown in the diagram. The motion of the particle is in a vertical plane containing a line of greatest slope of the inclined plane. \includegraphics[max width=\textwidth, alt={}, center]{3a1726d9-1b0c-41de-8b43-56019e18aac1-12_389_789_557_639}

1. Given that the time of flight from \(O\) to \(P\) is \(T\), find an expression for \(u\) in terms of \(\theta , \alpha , T\) and \(g\).
2. Using the identity \(\cos ( X - Y ) = \cos X \cos Y + \sin X \sin Y\), show that the distance \(O P\) is given by \(\frac { 2 u ^ { 2 } \sin \alpha \cos ( \alpha - \theta ) } { g \cos ^ { 2 } \theta }\).
   (6 marks)

10 A particle is projected up a long smooth slope at a speed of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The slope is at an angle \(\theta\) to the horizontal where \(\sin \theta = \frac { 1 } { 25 }\). After 2 seconds it passes a mark on the slope. Find the total time taken from the moment of projection until it passes the mark again and the total distance travelled in that time. {www.cie.org.uk} after the live examination series. }

A particle of mass 0.5 kg starts from rest and slides down a line of greatest slope of a smooth plane. The plane is inclined at an angle of 30° to the horizontal.

1. Find the time taken for the particle to reach a speed of 2.5 m s\(^{-1}\). [3]
2. Find the distance that the particle travels along the ground before it comes to rest. [3]

When the particle has travelled 3 m down the slope from its starting point, it reaches rough horizontal ground at the bottom of the slope. The frictional force acting on the particle is 1 N.

A car moves round a bend which is banked at a constant angle of \(10°\) to the horizontal. When the car is travelling at a constant speed of \(18 \text{ m s}^{-1}\), there is no sideways frictional force on the car. The car is modelled as a particle moving in a horizontal circle of radius \(r\) metres. Calculate the value of \(r\). [6]

A particle \(P\) is projected down a line of greatest slope on a smooth inclined plane. \(P\) has velocity \(5\text{ m s}^{-1}\) at the instant when it has been in motion for \(1.6\text{ s}\) and travelled a distance of \(6.4\text{ m}\). Calculate

1. the initial speed and the acceleration of \(P\), [5]
2. the inclination of the plane to the vertical. [3]

A car of mass \(m\) kg moves round a curve of radius \(r\) m on a road which is banked at an angle \(\theta\) to the horizontal. When the speed of the car is \(u\) ms\(^{-1}\), the car experiences no sideways frictional force. Given that \(\tan \theta = \frac{u^2}{gr}\), show that the sideways frictional force on the car when its speed is \(\frac{u}{2}\) ms\(^{-1}\) has magnitude \(\frac{3}{4}mg \sin \theta\) N. [10 marks]

A car moves round a circular racing track of radius 100 m, which is banked at an angle of 4° to the horizontal.

1. Show that when its speed is 8.28 ms\(^{-1}\), there is no sideways force acting on the car. [4 marks]
2. When the speed of the car is 12.5 ms\(^{-1}\), find the smallest value of the coefficient of friction between the car and the track which will prevent side-slip. [9 marks]

A cyclist travels on a banked track inclined at \(8°\) to the horizontal. He moves in a horizontal circle of radius 10 m at a constant speed of \(v\) ms\(^{-1}\). If there is no sideways frictional force on the cycle, calculate the value of \(v\). [6 marks]