Matthew Clemens, Physics and Math Teacher, Parent, Tutor, and Professional Ski Instructor
If we are assuming no air resistance and a vertical launch on Earth... The maximum height a projected ball will attain depends on the speed with which it is projected upward - also known as launch velocity. Now, the launch velocity depends on the mechanism projecting it upward and the mass of the object being projected. Things like the size of the force the mechanism can impart (which impacts how quickly it can accelerate during launch) and the distance through which the mechanism can impart the force are important, but it sounds like your question assumes these will be the same for each object. So, the issue now becomes how the mass of the object impacts things. In general, the more mass something has, the less quickly it can accelerate for a given Net Force. This is a function of Newton's 2nd Law: acceleration = Net Force / mass. So simply put, the larger the mass, the less quickly it will accelerate during launch, and therefore the lower its launch velocity will be... resulting in a lower maximum height. Additionally, there is the Net Force part of Newton's 2nd Law to consider, if we assume a launch with any vertical aspect to it. Mass will impact the weight, which is a force opposing the force the mechanism is exerting and therefore has an affect on the Net Force during launch. This results in an impact on acceleration during launch and therefore launch velocity. This is probably more than you were looking for, but hopefully it gives you some things to consider and look into a little more deeply. Sometimes an apparently "simple" question requires quite an in depth analysis. Enjoy digging deeper.