The intersection point of two moving 2D objects is equivalent to the intersection of a moving point with the Minkowski sum of the two objects -- the shape you get when you "inflate" one object by the other's shape. So, for instance, the intersection point of two moving circles of radius r is equivalent to the intersection of a moving point and a circle of 2r. If you reformulate the problem this way, you can sweep the moving point into a line segment (of length equal to the objects' relative movement) and find the exact intersection point.

Essentially this makes the intersection calculation easier at the expense of computing the Minkowski sum. However, computing the Minkowski sum is trivial in certain cases, and the axis-aligned square "grenade" intersecting with a line segment is one of them:

You can treat the grenade as a point and inflate the line segment by the grenade's dimensions in x and y to make a rectangle. Then you can sweep the point to make a line segment representing the grenade's movement and intersect it with the rectangle. This gives you the exact intersection point, much as in the case of the bullet/bin intersection, without needing to incrementally search along the grenade's path.

http://en.wikipedia.org/wiki/Minkowski_addition