OCR MEI M1 (Mechanics 1)

1 A girl throws a small stone with initial speed \(14 \mathrm {~ms} { } ^ { 1 }\) at an angle of \(60 ^ { \circ }\) to the horizontal from a point 1 m above the ground. She throws the stone directly towards a vertical wall of height 6 m standing on horizontal ground. The point O is on the ground directly below the point of projection, as shown in Fig. 8. Air resistance is negligible. \begin{figure}[h] \end{figure}

1. Write down an expression in terms of \(t\) for the horizontal displacement of the stone from O , \(t\) seconds after projection. Find also an expression for the height of the stone above O at this time. The stone is at the top of its trajectory when it passes over the wall.
2. (A) Find the time it takes for the stone to reach its highest point.
   (B) Calculate the distance of O from the base of the wall.
   (C) Show that the stone passes over the wall with 2.5 m clearance.
3. Find the cartesian equation of the trajectory of the stone referred to the horizontal and vertical axes, \(\mathrm { O } x\) and \(\mathrm { O } y\). There is no need to simplify your answer. The girl now moves away a further distance \(d \mathrm {~m}\) from the wall. She throws a stone as before and it just passes over the wall.
4. Calculate \(d\).

2 A particle is projected vertically upwards from a point O at \(21 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the greatest height reached by the particle. When this particle is at its highest point, a second particle is projected vertically upwards from \(O\) at \(15 \mathrm {~ms} ^ { - 1 }\).
2. Show that the particles collide 1.5 seconds later and determine the height above O at which the collision takes place.

3 The trajectory ABCD of a small stone moving with negligible air resistance is shown in Fig. 7. AD is horizontal and BC is parallel to AD . The stone is projected from A with speed \(40 \mathrm {~ms} ^ { - 1 }\) at \(50 ^ { \circ }\) to the horizontal. \begin{figure}[h] \end{figure}

1. Write down an expression for the horizontal displacement from A of the stone \(t\) seconds after projection. Write down also an expression for the vertical displacement at time \(t\).
2. Show that the stone takes 6.253 seconds (to three decimal places) to travel from A to D . Calculate the range of the stone. You are given that \(X = 30\).
3. Calculate the time it takes the stone to reach B . Hence determine the time for it to travel from A to C.
4. Calculate the direction of the motion of the stone at C .