Question

Can there be motion in two dimensions with an acceleration only in One dimension

Answer 1

For every vector defined as to betwo-dimensional, there is always a way to arrange your axis so it is defined by only one component. This case is only possible if the body has 2-d velocity at first, like the case of an anglede projectile. The acceleration will only add velocity to the the body in it's direction.

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Answer 2

Yes it is possible. The number of components a vector has depends on the choice of coordinate system. I can provide an example in which the velocity is two-dimensional and the acceleration is one dimensional.

Let’s imagine a body orbiting the sun in an elliptical orbit and let us choose a coordinate system in the plane of the orbit.

At most locations in the orbit, the velocity has two components. But at all times the direction of the acceleration is along a line towards the sun from the planet. So for a particular point in the orbit, let’s choose a coordinate system where the x axis is parallel to a line between the sun and the planet. In that coordinate system, the acceleration has only an x coordinate (one-dimensional). However the velocity of the planet, (for a non circular, elliptical orbit) is at an angle that is not perpendicular to the x chosen axis. Accordingly, the velocity has two components.

So there is your example.