The distance d through which a stone falls from rest is proportional to the square of the time taken t if the stone falls 45m in 3 seconds, how far will it fall in 6 seconds? How long will it take to fall 20 m?
We can express the relationship between distance and time as:
d = kt^2
where k is a constant of proportionality.
Given that the stone falls 45m in 3 seconds, we can substitute these values into the equation to solve for k:
45 = k * (3)^2
45 = 9k
k = 45/9
k = 5
Now, we can use this value of k to find the distance the stone will fall in 6 seconds:
d = 5 * (6)^2
d = 5 * 36
d = 180m
Therefore, the stone will fall 180m in 6 seconds.
To find out how long it will take the stone to fall 20m, we can rearrange the equation to solve for t:
d = kt^2
20 = 5 * t^2
4 = t^2
t = √(4)
t = 2 seconds
Therefore, it will take 2 seconds for the stone to fall 20m.
To solve this problem, we need to use the given information that the distance d is proportional to the square of the time taken t.
Step 1: Determine the constant of proportionality
Given that the stone falls 45m in 3 seconds, we can set up the following proportion:
d/t^2 = k
where k is the constant of proportionality.
Plugging in the values, we get:
45/3^2 = k
45/9 = k
k = 5
So, the constant of proportionality is 5.
Step 2: Calculate the distance fallen in 6 seconds
Now that we know the constant of proportionality, we can calculate the distance the stone will fall in 6 seconds.
Using the equation:
d = k * t^2
where k = 5 and t = 6, we have:
d = 5 * 6^2
d = 5 * 36
d = 180
Therefore, the stone will fall 180m in 6 seconds.
Step 3: Calculate the time taken to fall 20m
To determine the time it takes for the stone to fall 20m, we rearrange the equation:
d = k * t^2
to solve for t:
t^2 = d/k
Now, substitute the given values:
t^2 = 20/5
t^2 = 4
Taking the square root of both sides:
t = √4
t = 2
Therefore, it will take 2 seconds for the stone to fall 20m.