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Year 1 of the natural sciences course of the University of Cambridge

Mathematics course of the year 1 of the natural sciences course of the University of Cambridge

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge

2013

Paper 1

Rewrite the complex <position> as $z(t)=2+3i/(t-i)$. Differentiating gives the complex <velocity> and <acceleration>
$$
\dot z=-\frac{3i}{(t-i)^2},\qquad
\ddot z=\frac{6i}{(t-i)^3}.
$$
For real $t$, $|t-i|=\sqrt{t^2+1}$. Their requested magnitudes are therefore
$$
\boxed{|\dot z|=\frac3{t^2+1},\qquad
|\ddot z|=\frac6{(t^2+1)^{3/2}}.}
$$
There is no pole for real time. The magnitude is the Euclidean length of the two Cartesian components, not the modulus of either component separately.

Solution

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 1 / 12 / d / Solution

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 1 / 12 / d / Solution

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