Two plane $$ A $$ and $$ B $$ are flying al the same altitute, if their velocity a Va= $$ 500 \mathrm{~km} / \mathrm{hr} $$ and $$ V B=700 \mathrm{~km} / \mathrm{hr} $$. such that the angle between their straight line causes is $$ heta=60^{\circ} $$. Determine the valocity of plane $$ B $$ with bespect plane $$ A $$.

Question

Two plane $$ A $$ and $$ B $$ are flying al the same altitute, if their velocity a Va= $$ 500 \mathrm{~km} / \mathrm{hr} $$ and $$ V B=700 \mathrm{~km} / \mathrm{hr} $$. such that the angle between their straight line causes is $$ 	heta=60^{\circ} $$. Determine the valocity of plane $$ B $$ with bespect plane $$ A $$.

UpStudy AI · Accepted Answer

We are given that plane $$ A $$ has velocity
$$
V_A = 500\,\text{km/hr},
$$
and plane $$ B $$ has velocity
$$
V_B = 700\,\text{km/hr},
$$
with the angle between their paths being
$$
\theta = 60^\circ.
$$

The velocity of plane $$ B $$ relative to plane $$ A $$ is given by
$$
\vec{V}_{B/A} = \vec{V}_B - \vec{V}_A.
$$
The magnitude of the relative velocity is found using the Law of Cosines:
$$
\left|\vec{V}_{B/A}\right| = \sqrt{V_B^2 + V_A^2 - 2 V_A V_B \cos\theta}.
$$

Substitute the given values into the equation:
$$
\left|\vec{V}_{B/A}\right| = \sqrt{700^2 + 500^2 - 2 \times 700 \times 500 \cos 60^\circ}.
$$

Recall that
$$
\cos 60^\circ = \frac{1}{2}.
$$

Now, compute each term:
$$
700^2 = 490\,000,\quad 500^2 = 250\,000.
$$

The cross term becomes:
$$
2 \times 700 \times 500 \times \frac{1}{2} = 700 \times 500 = 350\,000.
$$

Thus, the magnitude is
$$
\left|\vec{V}_{B/A}\right| = \sqrt{490\,000 + 250\,000 - 350\,000} = \sqrt{740\,000 - 350\,000}.
$$
$$
\left|\vec{V}_{B/A}\right| = \sqrt{390\,000}.
$$

We can express this as:
$$
\left|\vec{V}_{B/A}\right| = 100 \sqrt{39}.
$$

Approximating $$\sqrt{39} \approx 6.245$$, we get:
$$
\left|\vec{V}_{B/A}\right| \approx 100 \times 6.245 = 624.5\,\text{km/hr}.
$$

Thus, the velocity of plane $$ B $$ with respect to plane $$ A $$ is approximately
$$
624.5\,\text{km/hr}.
$$
The velocity of plane $$ B $$ relative to plane $$ A $$ is approximately $$ 624.5 \, \text{km/hr} $$.

Two plane and are flying al
the same altitute, if their velocity a
Va= and .
such that the angle between their
straight line causes is .
Determine the valocity of plane
with bespect plane .

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Two plane and are flying al the same altitute, if their velocity a Va= and . such that the angle between their straight line causes is . Determine the valocity of plane with bespect plane .