1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta

A particle is projected from a point \(P\) on an inclined plane, up the line of greatest slope through \(P\), with initial speed \(V\). The angle of the plane to the horizontal is \(\theta\).

1. If the plane is smooth, and the particle travels for a time \(\frac{2V}{g}\cos\theta\) before coming instantaneously to rest, show that \(\theta = \frac{1}{4}\pi\). [4]
2. If the same plane is given a roughened surface, with a coefficient of friction 0.5, find the distance travelled before the particle comes instantaneously to rest. [5]

\includegraphics{figure_2} The diagram shows a sector \(OAB\) of a circle with centre \(O\) and radius \(r\) cm in which angle \(AOB\) is \(\theta\) radians. The sector has a perimeter of 18 cm.

1. Show that \(\theta = \frac{18 - 2r}{r}\). [2]
2. Find the area of the sector in terms of \(r\), simplifying your answer. [2]

\includegraphics{figure_11} A projectile is fired from a point \(O\) in a horizontal plane, with initial speed \(V\), at an angle \(\theta\) to the horizontal (see diagram).

1. Show that the range of the projectile on the horizontal plane is $$\frac{2V^2 \sin \theta \cos \theta}{g}.$$ [4]

There are two vertical walls, each of height \(h\), at distances 30 m and 70 m, respectively, from \(O\) with bases on the horizontal plane. The value of \(\theta\) is \(45°\).

1. If the projectile just clears both walls, state the range of the projectile. [1]
2. Hence find the value of \(V\) and of \(h\). [5]

\includegraphics{figure_9} The diagram shows a sector of a circle, \(OMN\). The angle \(MON\) is \(2x\) radians, the radius of the circle is \(r\) and \(O\) is the centre.

1. Find expressions, in terms of \(r\) and \(x\), for the area, \(A\), and perimeter, \(P\), of the sector. [2]
2. Given that \(P = 20\), show that \(A = \frac{100x}{(1 + x)^2}\). [2]
3. Find \(\frac{dA}{dx}\), and hence find the value of \(x\) for which the area of the sector is a maximum. [5]

\(P\) and \(Q\) are points on the circumference of a circle with centre \(O\) and radius \(r\). The angle \(POQ\) is \(\theta\) radians. Given that the chord \(PQ\) has length 4, find an expression for the length of the arc \(PQ\) in terms of \(\theta\) of only. [5]