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202403051417 Revision Paper XIII Q10

To a man walking along a horizontal road at 1.5\,\mathrm{m/s} the rain is coming towards him and appears to be falling at 3.5\,\mathrm{m/s} at an angle of 30^\circ to the vertical. Find, by calculation or drawing, the true speed of the rain and the angle this makes with the vertical.

Extracted from A. Godman & J. F. Talbert. (1973). Additional Mathematics Pure and Applied in SI Units.

Setup.

As always, visualise the scene.

Two observers in different reference frames S and S' can give different descriptions of the same physical event (x,y,z,t). Where something is depends on when you check on it and on the movement of your own reference frame. Time and space are not independent quantities; they are related by relative velocity.

If S' is moving with speed v in the positive x-direction relative to S, then its coordinates in S' are

\begin{aligned} x' & = x-vt \\ y' & = y \\ z' & = z \\ t' & = t \\ \end{aligned}

and if an object has velocity \mathbf{u} in frame S, then velocity \mathbf{u}' of the object in frame S' is

\begin{aligned} u' & = \frac{\mathrm{d}x'(t)}{\mathrm{d}t} \\ & = \frac{\mathrm{d}}{\mathrm{d}t}(x(t)-vt) \\ & = \frac{\mathrm{d}x(t)}{\mathrm{d}t}-v \\ & = u-v \\ \end{aligned}

in magnitude.

Cf. Ming-chang Chen. (2017). NTHU EE211000 Modern Physics.

Roughwork.

Recall that a triangle is uniquely determined by not all but some of its three sides and three angles:

\begin{aligned} \textrm{SSS} & \quad \textrm{(Side-Side-Side)} \\ \textrm{SAS} & \quad \textrm{(Side-Angle-Side)} \\ \textrm{ASA} & \quad \textrm{(Angle-Side-Angle)} \\ \textrm{AAS} & \quad \textrm{(Angle-Angle-Side)} \\ \textrm{RHS} & \quad \textrm{(Right angle-Hypotenus-Side)} \\ \end{aligned}

Hence, provided in part

\begin{aligned} u' & = 3.5 \\ v & = 1.5 \\ \measuredangle (u',v) & = 60^\circ \\ \end{aligned}

we can supply in whole

\begin{aligned} u & = \cdots \\ \measuredangle (v,u) & = \cdots \\ \measuredangle (u,u') & = \cdots \\ \end{aligned}

This problem is not to be attempted.