All Problems

Kinematics

Problem 1.24

A radius vector of a point \(A\) relative to the origin varies with time \(t\) as \(\mathbf{r}=a t \mathbf{i}-b t^{2} \mathbf{j},\) where \(a\) and \(b\) are positive constants, and \(\mathbf{i}\) and \(\mathbf{j}\) are the unit vectors of the \(x\) and \(y\) axes. Find: (a) the equation of the point's trajectory \(y(x) ;\) plot this function; (b) the time dependence of the velocity \(\mathbf{v}\) and acceleration \(\mathbf{w}\) vectors, as well as of the moduli of these quantities; (c) the time dependence of the angle \(\alpha\) between the vectors \(\mathbf{w}\) and \(\mathbf{v}\) (d) the mean velocity vector averaged over the first \(t\) seconds of motion, and the modulus of this vector.

Reveal Answer
 (a) y=−x2b/a2 (b) v=ai−2btj,w=−2bj,v==a2+4b2t2,w=2b (c) tan⁡α=a/2bt (d) ⟨v⟩=ai−btj,∣⟨v⟩∣==a2+b2t2\text { (a) } y=-x^{2} b / a^{2} \text { (b) } \quad \mathbf{v}=a \mathbf{i}-2 b t \mathbf{j}, \quad \mathbf{w}=-2 b \mathbf{j}, \quad v= =\sqrt{a^{2}+4 b^{2} t^{2},} w=2 b \text { (c) } \tan \alpha=a / 2 b t \text { (d) }\langle\mathbf{v}\rangle=a \mathbf{i}-b t \mathbf{j},|\langle\mathbf{v}\rangle|= =\sqrt{a^{2}+b^{2} t^{2}}
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