homework and exercises - What will be the trajectory of the given motion

If it is given that component of acceleration perpendicular to the velocity of a body has a constant, non-zero magnitude, how can we mathematically prove that the trajectory of the body will be circular?

I have basic knowledge of calculus, vectors and co-ordinate geometry. I can prove that the trajectory of a body in projectile motion is parabolic. I tried to follow a similar method to get an equation of circle but couldn't succeed. I tried forming parametric equations of x and y in terms of time t (the way we proceed when proving the parabolic path of a projectile) but could not reach the conclusion.

Edit: I have edited the question in order to improve it. However I still can not share my work on this question as I had asked the question almost 6 months ago and now I can not find where I tried solving it. Sincerely sorry for that.

Answer

Ok let's try. I think that there are a lot of way to do so and I will try one (maybe not the faster way, but should be clean enough). To understand the motion we need just two dimensions, so we work on a plane. We take a point and define its velocity $\vec{v}=(v_x,v_y)$ and say that it is subject to a constant (in module) acceleration that is orthogonal to $\vec{v}$, say $\vec{a}=(a_x,a_y)$. Let's fix an origin so that we can write

$$v_x = v\cos\theta \qquad v_y=v\sin\theta$$ where $\theta$ is an angle with respect to some origin point. Since $\vec{a}$ must be orthogonal, than it is in the form $$a_x = -a\sin\theta \qquad a_y=a\cos\theta$$ so that $\vec{v}\cdot\vec{a}=0$ (you can also exchange the signs, is the same). To find the trajectory we write the equations of motion of the components which are $$\frac{d v_x}{dt} = a_x \qquad \frac{d v_y}{dt} = a_y$$ Before starting doing derivatives, we can simplify the calculation proving that under our assumptions the module of the velocity $v$ is constant. Indeed $$\frac{dv^2}{dt}=\frac{d}{dt}\left(v_x^2+v_y^2\right) =2\left(v_x\frac{dv_x}{dt}+v_y\frac{dv_y}{dt}\right) =2\left(v_x a_x+v_y a_y\right) = 2\vec{v}\cdot\vec{a}=0$$ so we proved that a perpendicular acceleration cannot vary the module of the velocity but only rotate it. Now we substitute the expression of the components in the equations of motion, obtaining $$\frac{d v_x}{dt}=-v\sin\theta\frac{d\theta}{dt}=-a\sin\theta \qquad \frac{d v_y}{dt}=v\cos\theta\frac{d\theta}{dt}=a\cos\theta$$ so $$\frac{d\theta}{dt}=\frac{a}{v}$$ where the parameter on the right is a constant. Integrating $$\theta(t)=\frac{at}{v}+\phi$$ where it is nice to define $\omega=a/v$ and $\phi=\theta(0)$ which is just the initial angle at $t=0$. Then, putting it into the definitions of the velocities we have $$v_x = v\cos(\omega t+\phi) \qquad v_y = v\sin(\omega t+\phi)$$ Last step: integrate the positions. The coordinates are $$\frac{dx}{dt}=v_x \qquad \frac{dy}{dt}=v_y$$ and it is easy to integrate them obtaining $$x=x_0+\frac{v}{\omega}\sin(\omega t+\phi) \qquad y=y_0-\frac{v}{\omega}\cos(\omega t+\phi)$$ where $x_0$ and $y_0$ can be imposed by setting initial conditions. Notice that here you can recover the "standard" cosine-sine assignments just by adding a phase $\pi/2$ to $\phi$. Now, finally, we can recognize that this is a circular motion. Indeed, summing the square of the previous equations one finds that $$(x-x_0)^2+(y-y_0)^2=\left(\frac{v}{\omega}\right)^2$$ which is the circle equation with center in $(x_0,y_0)$ and radius $R=|v/\omega|$.