4. (18 pts) A 1.0 kg particle is released from rest at the origin \(\vec{\mathbf{r}}_{0} = (0 \hat{\mathbf{i}} + 0 \hat{\mathbf{j}}) \mathrm{m}\) , and is then subject to a position-dependent force given in Newtons by \[\vec{\mathbf{F}} (x,y) = (y \hat{\mathbf{i}} -x \hat{\mathbf{j}}).\] Furthermore, the particle is constrained to a parabolic curve so that it only moves along the path \[y = 10x - x^{2}\] where both \(x\) and \(y\) are measured in meters. (a) Is this force conservative? How can you tell? (b) Calculate the work done by the force on the particle as it moves along the parabolic path to a final position \(\vec{\mathbf{r}}_{f} = (10 \hat{\mathbf{i}} + 0 \hat{\mathbf{j}}) \mathrm{m}\) . (c) What is the speed of the particle after this displacement? 解题 讲题 注意 使用 MathTex 显示公式