Question

How to calculate radius of curvature in projectile motion?

Answer 1

When a particle moves along a curved path, it is continuously acted upon by centripetal force. Then only motion along a curve will be possible.

Now, trajectory of a projectile is parabola, a curved path.

The centripetal force at each point is provided by appropriate component of mg in the direction of radius of curvature at the point concerned. Here, m is mass of projectile and g is acceleration hue to gravity.

At the topmost point , the velocity of the projectile is ux, where ux is x component ( horizontal component) of initial velocity. Remember that ux remains constant throughout the motion.

Also, at the topmost point there is no y component of velocity.

So, at the top most point, the velocity is horizontal and hence, the radius of curvature at that point is vertically downward.

Now, general expression for centripetal force is mv^2/R, which in our case will be mux^2/R. Then,

mux^2/R=mg

Therefore, R=ux^2/g.

ux =u cos (theta). Her, ( theta) is angle of projection and u is velocity of projection

Answer 2

The formula for radius of curvature of a curve at any point is given as-

R=(1+(dy/dx)^(3/2))/(d2y/dx2)

To evaluate this you can find the trajectory of a projectile and then substitute the given first and second derivative to find the radius of curvature.