Question

A body of mass m is moving with a uniform speed V along an ellipse of semi major and semi minor axis of length a and b (as shown in the diagram). If the magnitude of change in velocity when the particle moves from point A to point B is V n , then find n . (Tangents to an ellipse at the ends of the axes are perpendicular to the respective axis)

Answer 1

If the magnitude of change in velocity when the particle moves from point A to point B is V n , then the value of n is √2

Explanation:

The figure is attached

The velocity at any point of the elliptical path will be tangential to that point (just like in circular motion)

Thus, the velocities at A and B will be in the direction shown

Given that the speed of the body of mass m is uniform and is equal to V

Thus, the velocity at point A will be

$\vec V_A=V\hat i$

And Velocity at point B will be

$\vec V_B=-V\hat j$

Change in velocity

$\Delta \vec V=\vec V_A-\vec V_B$

$\implies \Delta \vec V=V\hat i-(-V\hat j)$

$\implies \Delta \vec V=V\hat i+V\hat j$

Therefore, the magnitude of the change in velocity

$|\Delta \vec V|=\sqrt{V^2+V^2}$

$\implies |\Delta \vec V|=\sqrt{2}V$

Given that change in the magnitude of the velocity is nV

Therefore, comparing with the above we get

$n=\sqrt{2}$

Hope this answer is helpful.

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