Understanding the Physics

* Symmetry: When you throw a ball straight up, its upward journey takes the same amount of time as its downward journey. Since the total time is 4 seconds, the ball takes 2 seconds to go up and 2 seconds to come down.

* Acceleration due to Gravity: The ball experiences a constant downward acceleration due to gravity (approximately 9.8 m/s²).

Calculations

1. Downward Motion: We'll focus on the downward motion first.

* Initial velocity (v₀) = 0 m/s (the ball is momentarily at rest at the top of its trajectory)

* Time (t) = 2 s

* Acceleration (a) = 9.8 m/s²

2. Using the Equation: We can use the following kinematic equation to find the final velocity (v) of the ball just before it hits your hand:

v = v₀ + at

v = 0 + (9.8 m/s²)(2 s)

v = 19.6 m/s

3. Upward Motion: Since the upward motion is symmetrical, the initial upward velocity must be the same magnitude as the final downward velocity, but in the opposite direction. Therefore, the initial upward velocity is 19.6 m/s.

Answer: You must toss the ball straight up at a speed of 19.6 m/s for it to take 4 seconds to return to you.