# Acceleration vector at an acute angle to velocity vector

What will be the description of motion of particle if acceleration vector always make an acute angle with velocity vector?

I know that if acceleration vector makes right angle with velocity vector always then body will undergo uniform circular motion. but in this case what could be the resulting shape of path and how we can derive or prove it?

• "I know that if acceleration vector makes right angle with velocity vector always then body will undergo uniform circular motion." Not necessarily. Do you mean the special case where it is perpendicular and of magnitude $v^2/r$? Commented May 7, 2021 at 2:44

The acceleration can be broken down into two components, one parallel to the path and one orthogonal to it.

The orthogonal component causes the path to bend, the parallel component causes the particle to accelerate (forwards or backwards according to the component's direction).

The curvature of the bend depends on the speed of the particle; the faster it goes, the gentler the curve and vice versa.

So the particle travels in a spiral. Increasing speed creates an outward spiral, reducing speed creates an inward spiral.

• by " gentler the curve" you mean curvature increases? Commented May 6, 2021 at 11:52
• @lalittolani No, the opposite. A wide, gentle curve has a small curvature, a tight curl has a high curvature. Commented May 6, 2021 at 14:53
• ok really grateful to you Commented May 7, 2021 at 6:52