[In this question, the unit vectors $$\(\mathbf{i}\)$$ and $$\(\mathbf{j}\)$$ are in a vertical plane, with $$\(\mathbf{i}\)$$ being horizontal and $$\(\mathbf{j}\)$$ being vertically upwards.] A rough ramp is fixed to horizontal ground. The ramp is inclined to the ground at an angle $$\(\alpha\)$$, where $$\(\tan \alpha=\frac{7}{24}\)$$ The point $$\(A\)$$ is at the bottom of the ramp and the point $$\(B\)$$ is at the top of the ramp. The line $$\(A B\)$$ is a line of greatest slope of the ramp and $$\(A B=15 \mathrm{~m}\)$$, as shown in Figure 3. A particle $$\(P\)$$ of mass $$\(0.3 \mathrm{~kg}\)$$ is projected with speed $$\(U \mathrm{~ms}^{-1}\)$$ from $$\(A\)$$ directly towards $$\(B\)$$. At the instant $$\(P\)$$ reaches the point $$\(B\)$$, the velocity of $$\(P\)$$ is $$\((24 \mathbf{i}+7 \mathbf{j}) \mathrm{m} \mathrm{s}^{-1}\)$$ The particle leaves the ramp at $$\(B\)$$, and moves freely under gravity until it hits the horizontal ground at the point $$\(C\)$$. The coefficient of friction between $$\(P\)$$ and the ramp is $$\(\frac{1}{5}\)$$ (a) Find the work done against friction as $$\(P\)$$ moves from $$\(A\)$$ to $$\(B\)$$. (3) (b) Use the work-energy principle to find the value of $$\(U\)$$. (4) (c) Find the time taken by $$\(P\)$$ to move from $$\(B\)$$ to $$\(C\)$$. (3) At the instant immediately before $$\(P\)$$ hits the ground at $$\(C\)$$, the particle is moving downwards at $$\(\theta^{\circ}\)$$ to the horizontal. (d) Find the value of $$\(\theta\)$$ (4)

Answer:

Knowledge points:

Solution: