A wrecking ball is hanging at rest from a crane when suddenly the cable breaks. The time it takes for the ball to fall halfway to the ground is 4.36 s. Find the time it takes for the ball to fall from rest all the way to the ground.
I will be glad to critique your work.
Knowing that the distance fallen is proportional to t^2 should give you a very big hint.
You need to increase the time by a factor of sqrt2 = 1.414, at the half-way point.
To find the time it takes for the wrecking ball to fall all the way to the ground, we can use the concept of free fall and the equation for the time of flight.
Let's break down the problem:
Given:
- Time taken for the ball to fall halfway to the ground = 4.36 s.
We know that in the absence of air resistance, the time taken for an object to fall freely from rest to the ground is the same as the time taken for it to fall from the halfway point to the ground.
So, we can use the equation for the time of flight during free fall:
t = sqrt((2d) / g)
Where:
t = time of flight
d = distance fallen
g = acceleration due to gravity (approximately 9.8 m/s^2)
Since the ball falls halfway to the ground in 4.36 s, the distance fallen would be half of the total height.
Now, let's calculate the distance fallen:
We can use the equation for distance fallen during free fall:
d = 0.5 * g * t^2
Given:
t = 4.36 s
g = 9.8 m/s^2
Substituting the values:
d = 0.5 * 9.8 * (4.36)^2
d = 0.5 * 9.8 * 19.0096
d = 93.08496 m
Now, we know that the total distance from rest to the ground is twice the distance fallen halfway, so:
Total distance = 2 * 93.08496
Total distance = 186.16992 m
Finally, we can substitute the total distance into the equation for time of flight:
t = sqrt((2d) / g)
t = sqrt((2 * 186.16992) / 9.8)
t = sqrt(372.33984 / 9.8)
t = sqrt(38.0096)
t = 6.164 s
Therefore, it would take approximately 6.164 seconds for the wrecking ball to fall from rest all the way to the ground.