M2 January 2006 Q7

EdexcelOld spec14 marksProjectiles

7.

Figure 3: ball projected from A, 1 m above the ground, at 11 m/s at 30 degrees, hitting a pole of height 2 m at C below the target T, the pole 10 m horizontally from A
Figure 3

The object of a game is to throw a ball \(B\) from a point \(A\) to hit a target \(T\) which is placed at the top of a vertical pole, as shown in Figure 3. The point \(A\) is 1 m above horizontal ground and the height of the pole is 2 m. The pole is at a horizontal distance of 10 m from \(A\). The ball \(B\) is projected from \(A\) with a speed of 11 m s\(^{-1}\) at an angle of elevation of 30\(^\circ\). The ball hits the pole at the point \(C\). The ball \(B\) and the target \(T\) are modelled as particles.

(a) Calculate, to 2 decimal places, the time taken for \(B\) to move from \(A\) to \(C\). (3)
(b) Show that \(C\) is approximately 0.63 m below \(T\). (4)

The ball is thrown again from \(A\). The speed of projection of \(B\) is increased to \(V\) m s\(^{-1}\), the angle of elevation remaining 30\(^\circ\). This time \(B\) hits \(T\).

(c) Calculate the value of \(V\). (6)
(d) Explain why, in practice, a range of values of \(V\) would result in \(B\) hitting the target. (1)
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