Two girls decide to jump off a diving board. Katherine steps off the diving board. Anna runs straight off the diving board so that her initial velocity is solely horizontal. They both leave the diving board at the same time. Which one lands in the water first?

Anna, Katherine, or at the same time?
I think its at the same time but I just want to double check!

yes, again -g the whole time. Horizontal direction is another problem entirely.

To determine which girl lands in the water first, we need to consider the two factors that may affect their descent: vertical velocity and horizontal velocity.

Let's start with Katherine. Since she steps off the diving board, her initial vertical velocity will be zero. However, due to gravity, she will accelerate downward at a rate of approximately 9.8 meters per second squared. This means that her vertical velocity will continuously increase as she falls towards the water. On the other hand, Katherine has no horizontal velocity as she steps off the diving board.

Now, let's analyze Anna. Since she runs straight off the diving board, her initial horizontal velocity will be non-zero. However, since there is no initial vertical velocity, Anna will experience the same downward acceleration due to gravity as Katherine. Hence, Anna's vertical velocity will also increase over time.

Now, here comes the crucial part. Since the vertical motion of each girl is independent of their horizontal motion, the time taken for Katherine and Anna to reach the water will only depend on their vertical velocities. The horizontal motion does not affect the descent time in this scenario.

Since both Katherine and Anna start with the same vertical velocity of zero and experience the same downward acceleration due to gravity, their descent times will be equal. Therefore, Katherine and Anna will land in the water at the same time.

So, your initial intuition was correct! Both Katherine and Anna will land in the water at the same time when they jump off the diving board.