All Problems

Kinematics

Problem 1.25

A point moves in the plane \(x y\) according to the law \(x=a t\), \(y=\) at \((1 \stackrel{-\alpha t}{t}),\) where \(a\) and \(\alpha\) are positive constants, and \(t\) is time. Find: (a) the equation of the point's trajectory \(y(x)\); plot this function; (b) the velocity \(v\) and the acceleration \(w\) of the point as functions of time; (c) the moment \(t_{0}\) at which the velocity vector forms an angle \(\pi / 4\) with the acceleration vector.

Reveal Answer
 (a) y=x−x2α/a (b) v=a1+(1−2αt)2,w=2αa== const;   (c) t0=1/α\text { (a) } y=x-x^{2} \alpha / a \text { (b) } v=a \sqrt{1+(1-2 \alpha t)^{2}}, w=2 \alpha a= \begin{aligned} &=\text { const; }\ &\text { (c) } t_{0}=1 / \alpha \end{aligned}
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