# Direction of force when ball strikes an edge

From the perspective of classical mechanics, when a ball strikes an edge of a box-like-object, as shown here what is the direction of the force from the ball on the object (assuming there is no friction)? Is it always along the line joining the centre of the ball to the point of impact (assuming the ball is a uniform sphere)?

Both Ben Crowell's and sammy gerbil's answers are correct, and they highlight the shortcomings of idealized rigid-body physics models.

In the context of a short homework question, Ben Crowell's answer is probably the one you want, and the accepted answer to the question sammy gerbil linked in his comment will help out with the math. (Although you'll have to translate it from pseudo-code programming style.)

The only thing I'd add is that this idealized model is probably useless except maybe for video-game design. A frictionless rigid-body model works OK for some collisions of smooth, highly elastic objects, but I think you'd have trouble finding objects smooth enough and ridged enough for this approximation of a collision with a sharp edge to be suitable. In particular, if the ball grips against, deforms around, or is chipped by the edge at all, then in addition to the energy lost by that action (damage), it will gain some rotational momentum, which would affect its post-collision velocity vector.

Usually when two objects collide they make contact along a common surface plane. It might be a very small area, almost a point, but the direction of the normal at this point can be identified without ambiguity. The normal contact force between the object lies along the normal to this contact plane at the point of contact.

For ideal edges the direction of the force is mathematically indeterminate. This is because the surface changes direction discontinuously along an infinitesimally thin line. On one side of the line the surface normal points in one direction, on the other side of the line it points in a different direction. Exactly at the corner it is ambiguous which direction the surface normal is pointing. The discontinuity exists regardless of how close you get to the line.

Practically this might not be a problem for a point object, because if you look on a small enough scale you can always decide on which side of the line the point of contact lies, and therefore in which direction the normal contact force acts.

Real edges (also corners) are rounded to some extent. The local surface normal changes smoothly from one direction to another. If you look on a small enough scale you will be able to identify a local surface plane and its normal. Then you can decide how a point object will bounce if it strikes this local plane.

Real objects have finite size. If they are perfectly rigid then a single point of contact can be identified, and a local surface normal, along which the normal force is exerted during a collision. The direction of friction (if any) can also be determined.

However, if the object is finite and also deformable then it will bend around the edge or corner as it collides, making contact over a curved surface which has normals pointing in a range of directions. It is a very difficult problem to work out in what direction the resultant force on the object will act. Small differences in the line along which the ball approaches, or its angle of approach, or its orientation or speed and direction of rotation, or its elasticity, etc - all could make a significant difference to the direction in which the object will bounce off the edge or corner.

Is it always along the line joining the centre of the ball to the point of impact (assuming the ball is a uniform sphere)?

Yes, if there is no friction. The normal force is always along this line, because the normal force is always perpendicular to the surface of contact. The surface of contact is tangent to the surface of the ball, and if we imagine that the edge of the box is slightly rounded (which it has to be at the microscopic scale), then this would also hold for the box.