To find the time it takes for the ball to hit the ground, we can use the equation of motion for vertical motion:
s = ut + (1/2)at²
Where: s is the displacement (in this case, -152.4 m as the ball is moving downwards), u is the initial velocity (4 m/s, upwards), t is the time, and a is the acceleration (which is -9.8 m/s², as the ball is under the influence of gravity).
Plugging in the known values, we get:
-152.4 = 4t + (1/2)(-9.8)t²
Now, let's simplify and solve the equation:
-152.4 = 4t - 4.9t²
Rearranging the equation to bring all terms to one side, we have:
4.9t² - 4t - 152.4 = 0
To solve this quadratic equation, we can use the quadratic formula:
t = (-b ± √(b² - 4ac)) / (2a)
For our equation, a = 4.9, b = -4, and c = -152.4. Plugging in these values, we have:
t = (-(-4) ± √((-4)² - 4(4.9)(-152.4))) / (2(4.9))
Simplifying further:
t = (4 ± √(16 + 2998.56)) / 9.8
t = (4 ± √3014.56) / 9.8
Now, we have two possible solutions:
t₁ = (4 + √3014.56) / 9.8
t₂ = (4 - √3014.56) / 9.8
Calculating these values:
t₁ ≈ 7.67 seconds t₂ ≈ -2.43 seconds
Since time cannot be negative in this context, we disregard the negative value. Therefore, the time it takes for the ball to hit the ground is approximately 7.67 seconds.