two rocks are dropped off a cliff. the second rock hits 1.6 s after the first. how far below the cliff is the second stone when the rocks are 36m apart? (10.97m)
To find the distance below the cliff that the second stone is when the rocks are 36 m apart, we can use the formula for distance fallen by a freely falling object:
d = (1/2) * g * t^2
Where:
d is the distance fallen,
g is the acceleration due to gravity (approximately 9.8 m/s^2),
and t is the time taken.
Let's break down the problem step by step:
Step 1: Find the time taken by the second rock to fall.
Given that the second rock hits 1.6 s after the first, we can represent the time taken by the second rock as follows:
t2 = t1 + 1.6
Where:
t2 is the time taken by the second rock,
and t1 is the time taken by the first rock.
Step 2: Find the distance fallen by the first rock.
Since both rocks are dropped from the same cliff, the distance fallen by the first rock is given by:
d1 = (1/2) * g * t1^2
Step 3: Find the distance fallen by the second rock.
The distance fallen by the second rock is given by:
d2 = (1/2) * g * t2^2
Step 4: Find the distance between the rocks.
The distance between the rocks is given as 36 m, so:
d2 - d1 = 36
Now, let's substitute the known values and solve the equations:
d2 = (1/2) * g * (t1 + 1.6)^2
d1 = (1/2) * g * t1^2
d2 - d1 = 36
The given answer is 10.97 m, so we need to confirm if it satisfies the equations.
Let's calculate:
d2 - d1 = (1/2) * g * (t1 + 1.6)^2 - (1/2) * g * t1^2
Substituting the given values:
10.97 = (1/2) * 9.8 * (t1 + 1.6)^2 - (1/2) * 9.8 * t1^2
Now, we can solve this equation for t1:
10.97 = 4.9 * (t1^2 + 3.2t1 + 2.56) - 4.9t1^2
Simplifying:
10.97 = 4.9t1^2 + 15.68t1 + 12.704 - 4.9t1^2
Combining like terms:
10.97 = 15.68t1 + 12.704
Rearranging the equation:
15.68t1 = 10.97 - 12.704
15.68t1 = -1.734
t1 = -1.734 / 15.68
t1 ≈ -0.11 s
Since time cannot be negative, this result is not valid. Therefore, the given answer of 10.97 m does not satisfy the equations.
To find the distance below the cliff where the second stone is when the rocks are 36 meters apart, we can first calculate the time it took for the second stone to hit after the first stone.
We know that the second rock hits 1.6 seconds after the first rock. This means that the time for the first rock to hit is 0 seconds since it was dropped initially.
Using the equation of motion for freely falling objects, we have:
distance = (initial velocity × time) + (0.5 × acceleration × time²)
Since the rocks are dropped, the initial velocity for both rocks is zero.
For the first rock:
distance₁ = 0.5 × acceleration × (time for the first rock)²
For the second rock:
distance₂ = 0.5 × acceleration × (time for the second rock)²
We also know that the distance between the two rocks is 36 meters:
distance₁ - distance₂ = 36
Simplifying and rearranging the equations, we have:
0.5 × acceleration × (time for the first rock)² - 0.5 × acceleration × (time for the second rock)² = 36
Since the acceleration due to gravity is roughly 9.8 m/s² on Earth, we substitute this value into the equation:
0.5 × 9.8 × (time for the first rock)² - 0.5 × 9.8 × (time for the second rock)² = 36
Now, we substitute the given time difference of 1.6 seconds:
0.5 × 9.8 × (0)² - 0.5 × 9.8 × (1.6)² = 36
Simplifying further, we get:
0 - 0.5 × 9.8 × 2.56 = 36
-0.5 × 9.8 × 2.56 = 36
Now we can solve for the unknown variable (the distance below the cliff where the second stone is located) by dividing both sides by -12.32:
distance₂ = 36 / -12.32
distance₂ ≈ -2.92 m
Therefore, the second stone is approximately 2.92 meters below the cliff when the rocks are 36 meters apart.