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babunello [35]
3 years ago
10

If position vector r = bt^2i + ct^3j, where b and c are positive constants, when does the velocity vector make an angle of 450 w

ith the x-and y-axes?
Physics
1 answer:
vazorg [7]3 years ago
7 0

\vec r(t)=bt^2\,\vec\imath+ct^3\,\vec\jmath

The velocity at time t is

\dfrac{\mathrm d\vec r(t)}{\mathrm dt}=2bt\,\vec\imath+3ct^2\,\vec\jmath

Take two vectors that point in the positive x and positive y directions, such as \vec\imath and \vec\jmath. The dot products of the velocity vector with \vec\imath and \vec\jmath are

\dfrac{\mathrm d\vec r(t)}{\mathrm dt}\cdot\vec\imath=2bt=\sqrt{4b^2t^2+9c^2t^4}\cos\theta

and

\dfrac{\mathrm d\vec r(t)}{\mathrm dt}\cdot\vec\jmath=3ct^2=\sqrt{4b^2t^2+9c^2t^4}\cos\theta

We want the angles between these vectors to be 45º, for which we have \cos45^\circ=\frac1{\sqrt2}. So

\begin{cases}2\sqrt2\,bt=\sqrt{4b^2t^2+9c^2t^4}\\3\sqrt2\,ct^2=\sqrt{4b^2t^2+9c^2t^4}\end{cases}\implies3\sqrt2\,ct^2-2\sqrt2\,bt=0

\implies t(3ct-2b)=0

\implies t=0\text{ or }t=\dfrac{2b}{3c}

When t=0, the velocity vector is equal to the zero vector, which technically has no direction/doesn't make an angle with any other vector. So the only time this happens is for

\boxed{t=\dfrac{2b}{3c}}

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