M2 January 2011 Q6

EdexcelOld spec12 marksProjectiles

6. [In this question, the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in a vertical plane, \(\mathbf{i}\) being horizontal and \(\mathbf{j}\) being vertically upwards.]

Figure 3: particle projected from A, 10 m above O, with velocity (3i + 5j) m/s, landing on the horizontal ground
Figure 3

At time \(t = 0\), a particle \(P\) is projected from the point \(A\) which has position vector \(10\mathbf{j}\) metres with respect to a fixed origin \(O\) at ground level. The ground is horizontal. The velocity of projection of \(P\) is \((3\mathbf{i} + 5\mathbf{j})\) m s\(^{-1}\), as shown in Figure 3. The particle moves freely under gravity and reaches the ground after \(T\) seconds.

(a) For \(0 \leqslant t \leqslant T\), show that, with respect to \(O\), the position vector, \(\mathbf{r}\) metres, of \(P\) at time \(t\) seconds is given by \[\mathbf{r} = 3t\mathbf{i} + (10 + 5t - 4.9t^2)\mathbf{j}\] (3)
(b) Find the value of \(T\). (3)
(c) Find the velocity of \(P\) at time \(t\) seconds \((0 \leqslant t \leqslant T)\). (2)

When \(P\) is at the point \(B\), the direction of motion of \(P\) is 45\(^\circ\) below the horizontal.

(d) Find the time taken for \(P\) to move from \(A\) to \(B\). (2)
(e) Find the speed of \(P\) as it passes through \(B\). (2)
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