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Kinematics

Problem 1.29

A body is thrown from the surface of the Earth at an angle \(\alpha\) to the horizontal with the initial velocity \(v_{0}\). Assuming the air drag to be negligible, find: (a) the time of motion; (b) the maximum height of ascent and the horizontal range; at what value of the angle \(\alpha\) they will be equal to each other; (c) the equation of trajectory \(y(x)\), where \(y\) and \(x\) are displacements of the body along the vertical and the horizontal respectively; (d) the curvature radii of trajectory at its initial point and at its peak.

Reveal Answer
 (a) τ=2(v0/g)sin⁡α; (b) h=(v02/2g)sin⁡2α,l=(v02/g)sin⁡2α α=76∘ (c) y=xtan⁡α−(g/2v02cos⁡2α)x2 (d) R1=v02/gcos⁡α,R2=(v02/g)cos⁡2α\text { (a) } \tau=2\left(v_{0} / g\right) \sin \alpha;\begin{aligned} &\text { (b) } h=\left(v_{0}^{2} / 2 g\right) \sin ^{2} \alpha, l=\left(v_{0}^{2} / g\right) \sin 2 \alpha\ &\alpha=76^{\circ} \end{aligned} \text { (c) } y=x \tan \alpha-\left(g / 2 v_{0}^{2} \cos ^{2} \alpha\right) x^{2} \text { (d) } R_{1}=v_{0}^{2} / g \cos \alpha, R_{2}=\left(v_{0}^{2} / g\right) \cos ^{2} \alpha
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