Think about this. You're always travelling through spacetime with a constant speed (the speed of causality == the speed of light). This is your 4 vector because it has 4 components: x, y, z, t. Therefore the faster you go in one dimension (say, x), the slower you go in the other dimensions. The dimension that's affected the most by the warping of spacetime is time (t) in most cases, because you're moving much slower in the other 3. As you go faster your path becomes less warped. This is why the faster your ball is, the less curved its path look like. This also explains why objects with zero x/y/z speed fall straight towards the object that causes the warping: only t remains from your 4 vector so you're essentially moving through spacetime with the speed of causality (light) --> you fall straight down.