A Level Maths Question (1)

A particle P is initially at the point (2,6) in relation to an origin O. It is moving with velocity (3i + j) and constant acceleration (16i + 24j) , show that in 2 seconds, it is moving directly away from O.

- for this question my initial thought was just to find the final velocity and then prove that V is a multiple of U i.e its parallel and since it starts from the same point its moved in the same direction. But V= 35i +49J so my proof wont work.
1) what's wrong with the logic I've used ? a particle has to travel in the same direction as it's velocity no ?
2) what's a better/ correct solution ?

If its moving directly away from the origin at a point in time, then the current velocity v must be a positive scalar multiple of the position.

A sketch of the postiion (as a point) and the velocity as an arrow with the tail at the point/current position should make it clear?

oh that makes sense thank you! but why isn't the velocity being a positive scalar multiple of the initial velocity proof for it travelling in the same direction is that wrong to assume ?

You could "calculate" the positions and velocities at t=0 and t=2 and sketch them. There isnt really a relationship between them in the way you seem to be thinking. You integrate acceleration (twice) to get the position curve which joins the two points (and beyond for t>2) and you imagine the current velocity arrow (at time t) as being tangential to the positiion curve at time (t) as in the previous post.

So something like
https://www.statisticshowto.com/tangent-vector-velocity/
where T is the velocity vector at each point (position/time). The first point on the curve (assuming it starts at O) is something like this question is after.

If its moving directly away from the origin at a point in time, then the current velocity v must be a positive scalar multiple of the position.

A sketch of the postiion (as a point) and the velocity as an arrow with the tail at the point/current position should make it clear?

Sorry to bother you again with this but what would it mean if the velocity wasn’t multiple of displacement? Is it possible? Would it it just not pass through the origin ?