# How would I calculate the velocity of a sphere with an impulse applied at the top?

I'm developing a rigid body physics engine and currently have accurately calculated the angular velocity of the sphere when impulsed. Unfortunately though, I'm not calculating the velocity correctly as I'm struggling to understand what should happen. Say for example the sphere has an impulse applied at the top, not only would it move forwards and downwards, but it would also rotate forwards and curve downwards, here's a diagram of what I'm explaining:

Basically, S represents the sphere, the blue arrow represents the angular velocity, and black is the velocity I think the ball will obtain after the impulse as the ball is hit off centre at the top with a velocity curving downwards and the red is the velocity without the curve (not accurate just a diagram) I know that if the sphere had an impulse applied directly on the centre of the sphere it wouldn't gain any angular velocity nor would it curve downwards, this hypothetically the scenario where gravity wouldn't take effect.

My question then is A) would the ball curve downwards along the black arrow as I described? or B) would it follow a linear path even if it did angularly rotate?

How would I go about calculating the velocity of the ball if this impulse was applied at the top as I've described?

• Remember Newton's Third Law. If the ball is deflected downward, then it must be exerting a similar push against something else that moves upward, to maintain conservation of momentum. So, the question is: what is that other thing it's interacting with to generate a downward push? Are you trying to simulate friction with an atmosphere, or a table on which the ball is resting (as though viewed top-down like a billiards simulation)? Or, if this ball exists in a vacuum with nothing around it, then there might be no downward deflection at all. Apr 29, 2022 at 16:10
• Currently the other thing isn't anything, it's just a method to apply the impulse to the top of the sphere, there is no atmosphere or any other force applying to the sphere, just the sphere itself. I understand in an atmosphere, the magnus effect would take place, yet in this situation, since there's no atmosphere, the magnus effect wouldn't apply. Would the spheres velocity linearly move forwards and down or would it curve?
– Joe
Apr 29, 2022 at 16:18
• To curve, there would need to be a downward acceleration applied to the ball, meaning there would need to be an upward acceleration applied to something else - an equal and opposite reaction. If you don't have a "something else" to push up on, then there's nothing to push down on the ball, and no resulting curve. Apr 29, 2022 at 16:40

Think of this in terms of conservation of momentum / action and reaction. Let's work in an inertial frame that's co-moving with the ball, so that in this coordinate system, the ball is initially stationary at the origin.

We can imagine the impulse you applied as being the recoil from a tiny little gun mounted on the rim of the ball, shooting a bullet to the left. The exploding gunpowder applies a momentary impulse to the surface of the ball, pointing right, and to the bullet, pointing left.

The bullet flies away with some momentum in the leftward direction. Since we started with zero momentum in the system, the ball's resulting momentum vector must be exactly the negation of this, in order to conserve the total momentum of the system at zero. But since the bullet is flying directly to the left, it has no up or down component. That means the ball's momentum must also have no up or down component (there's no corresponding component in the bullet's momentum to neutralize it).

So, your intuition that the ball's movement should be angled downward is incorrect, at least for a ball floating in a frictionless vacuum. Any curving or diagonal deflection we observe in real-world examples would be due to a combination of factors:

• friction with the table the ball is rolling on

• friction with the air around the ball (the Magnus effect)

• the initial impulse not actually pointing directly to the left (eg. another ball or a cue stick colliding with the ball would apply an impulse with a component along the collision normal)

You also can't ever get a curving path from just one initial impulse. To get a curve, there must be something acting continuously on the ball throughout its travel, to continue deflecting the momentum downward (and correspondingly, something else in the system should get accelerated upward to balance out the reaction force). Without any external forces/impulses acting on a body, its center of mass must continue moving in a straight line of constant velocity.