Problem 7.40

îĵ

พิจารณาการเคลื่อนที่ในช่วงที่มีความเร่ง \(\vec{a} = acosθ + asinθ\)

จาก\(\vec{s}=\vec{u}t+\frac{1}{2}\vec{a}t^{2}\)

îĵîĵ

\(x_{1}+y_{1}=(0)t+\frac{1}{2}(acos\theta +asin\theta )t^{2}\)

îĵîĵ

\(x_{1} + y_{1} =\frac{at^{2}\cos \theta}{2}+\frac{at^{2}\sin \theta}{2}\)

\(\therefore y_{1} =\frac{at^{2}\sin \theta}{2}\)

จาก \(\vec{v} = \vec{u} + \vec{a}t\)

îĵ

\(\vec{v}_{1} = (0) + (acos\theta + asin\theta )t\)

îĵ

\(\vec{v}_{1} = atcos\theta + atsin\theta\)

\(|\vec{v}_{1}|^{2} = a^{2}t^{2}\cos ^{2}\theta + a^{2}t^{2}\sin ^{2}\theta = a^{2}t^{2}\)

พิจารณาการเคลื่อนที่หลังจากเชื้อเพลิงหมด

จาก \(v^{2} = u^{2} + 2\vec{a}\cdot \vec{s}\)

ĵîĵ

\(a^{2}t^{2}\cos ^{2}\theta = v_{1}^{2} + 2(-g )\cdot (x_{2} + y_{2} )\)

\(y_{2} =\frac{v_{1}^{2} - a^{2}t^{2}\cos ^{2}\theta}{2g}=\frac{a^{2}t^{2}(1 - \cos ^{2}\theta )}{2g}=\frac{a^{2}t^{2}\sin ^{2}\theta}{2g}\)

\(\therefore y = y_{1} + y_{2} =\frac{at^{2}\sin \theta}{2}+\frac{a^{2}t^{2}\sin ^{2}\theta}{2g}=\frac{at^{2}\sin \theta (g + asin\theta )}{2g}\;\blacksquare\)