Tangential acceleration in elliptical orbit?

[erisedk]

Homework Statement

A satellite S is moving in an elliptical orbit around the earth. The mass of the satellite is very small compared to the mass of the earth. Choose the correct statement.
Ans:
(A) the acceleration of S is always directed towards the centre of the earth

Homework Equations

F= GMm/R²

The Attempt at a Solution

In an elliptical orbit, since the velocity changes at different points, shouldn't there be a tangential acceleration as well, which would alter the direction of the net acceleration (tangential + radial)?
I know however, that there isn't a tangential acceleration, because till date, I have never considered it in any problem. I just can't justify why it wouldn't exist. Please help?

 

[gneill]

Accelerations are caused by forces.

Do the usual: Draw a free body diagram and identify the sources of all forces acting on the satellite. What do you find?

 

[erisedk]

Only the gravitational force. Does this somehow invalidate the formula tangential acceleration $a_{t} = \dfrac{d\vec{v}}{dt}$?

 

[PeroK]

Only the gravitational force. Does this somehow invalidate the formula tangential acceleration $a_{t} = \dfrac{d\vec{v}}{dt}$?

You're thinking of:

$a_t = \dfrac{dv}{dt}$

That's the component of the accleration in the direction of the instantaneous velocity. That need not be zero, as there is a component of acceleration normal to the velocity. And the (vector) sum of the two will be towards the focus.

 

[erisedk]

The centripetal acceleration is not towards the focus? Is it towards the centre of the instantaneous radius of curvature of the part of the ellipse where the satellite is present?

 

[erisedk]

Oh yeah you said it is normal to the instantaneous velocity. So, centripetal acceleration isn't towards the focus. Got it. The resultant acceleration is towards the focus. Thank you!

 

[PeroK]

Oh yeah you said it is normal to the instantaneous velocity. So, centripetal acceleration isn't towards the focus. Got it. The resultant acceleration is towards the focus. Thank you!

"Centripetal" literally means "towards the centre", so it's not quite appropriate for an elliptical orbit. If the orbit is circular, then the speed of the satellite must be constant. But, for an elliptical orbit, the speed will only be instantenously constant when the satellite is moving perpendicular to the line between it and the planet; at other times, the speed must be changing - as the acceleration cannot be normal to the instantaneous velocity.

 

[erisedk]

"Centripetal" literally means "towards the centre", so it's not quite appropriate for an elliptical orbit. If the orbit is circular, then the speed of the satellite must be constant. But, for an elliptical orbit, the speed will only be instantenously constant when the satellite is moving perpendicular to the line between it and the planet; at other times, the speed must be changing - as the acceleration cannot be normal to the instantaneous velocity.

Why did you say this? I mean I agree with it, but what did I say wrong?

 

[PeroK]

Why did you say this? I mean I agree with it, but what did I say wrong?

Normally centripetal acceleration is towards a fixed centre. You have the centripetal and angular components. Alternatively, you have the components of acceleration tangential and normal to the instantaneous velocity: neither of these is centripetal.

It's a minor point, but I think you should have said "the normal component isn't towards the focus".

 

[erisedk]

Ok thank you! :)