# Find the angle between the velocity and the total acceleration

A point moves along the arc of a circle of radius R .It's velocity depends upon the distance covered as v=a*s^1/2 where a is a constant .Find the angle between the velocity and the total acceleration vector as a function of distance.

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The solution turns out to be a first order differenetial equation

Since V = a*sqrt(s) where s is the distance traveled

we can relate s to the radius and the angle subtended

s = r*theta (just basic geometry here)

so V = a*sqrt(r*theta)

From basic kinematics

Thus we have

r*dtheta/dt = a*sqrt(r*theta)

By using separation of variable it is possible to solve for theta as an explicit function of time. In this case the initial condition is going to be the initial angle which you can assume is zero as it is simply a reference loctation.

If you want to know theta as a function of distance, then this really isnt much of a problem because theta = distance/radius

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Abercrombie, the angle to be determined in the OP's post is the angle between net acceleration and velocity of the object, and not the angle subtended at the centre as a function of time.