To determine the time it takes for a ball to return to its starting point when thrown vertically upwards, we can use the equation of motion for vertical motion under constant acceleration. In this case, the acceleration is due to gravity and is equal to approximately 9.8 m/s² (assuming we neglect air resistance).
The equation we can use is:
h = ut + (1/2)gt²
Where: h = maximum height reached by the ball (in this case, it will be zero when it returns to the starting point) u = initial velocity (30 m/s, upwards) g = acceleration due to gravity (9.8 m/s²) t = time
Rearranging the equation, we get:
0 = ut - (1/2)gt²
Since we want to find the time it takes for the ball to return to its starting point, we can solve for t. Let's substitute the given values and solve for t:
0 = (30 m/s)t - (1/2)(9.8 m/s²)t²
0 = 30t - 4.9t²
Rearranging further:
4.9t² - 30t = 0
We can factor out t:
t(4.9t - 30) = 0
This equation will be true when either t = 0 or 4.9t - 30 = 0. Since t = 0 corresponds to the initial time, we discard it in this case.
Solving 4.9t - 30 = 0 for t:
4.9t = 30
t = 30 / 4.9
t ≈ 6.1224 seconds
Therefore, it will take approximately 6.1224 seconds for the ball to return to its starting point when thrown vertically upwards with a velocity of 30 m/s.