A rock is dropped from a tree of height 20 m into a lake (depth 4.6 m) below. After entering the water, the rock floats gently down through the water at a constant speed of 1.6 m/s to the bottom of the lake. What is the total elapsed time?
To find the total elapsed time, we need to calculate the time it takes for the rock to fall from the tree to the surface of the water and the time it takes to sink from the surface of the water to the bottom of the lake.
1. Time to fall from the tree to the surface of the water:
Using the equation of motion, h = (1/2) * g * t^2, where h is the height, g is the acceleration due to gravity (approximately 9.8 m/s^2), and t is the time.
Rearranging the equation, t = sqrt(2h/g).
Substituting the given values, t = sqrt(2 * 20 / 9.8) = sqrt(40/9.8) ≈ 2.02 seconds.
2. Time to sink from the surface of the water to the bottom of the lake:
Since the rock sinks at a constant speed of 1.6 m/s, the time is equal to the distance divided by the speed.
The distance is the depth of the lake, which is 4.6 m.
The time is therefore 4.6 / 1.6 ≈ 2.88 seconds.
3. Adding the times together, the total elapsed time is approximately 2.02 + 2.88 = 4.9 seconds.
To find the total elapsed time, we need to calculate the time it takes for the rock to fall from the tree and the time it takes for the rock to sink to the bottom of the lake.
First, let's find the time it takes for the rock to fall from the tree. We can use the equation for the time it takes for an object to fall vertically:
t = √(2h/g)
Where:
t = time
h = height the object is dropped from
g = acceleration due to gravity (approximately 9.8 m/s^2)
Plugging in the values, we have:
t = √(2 * 20 m / 9.8 m/s^2)
t = √(40 / 9.8)
t ≈ 2.02 s
So, it takes approximately 2.02 seconds for the rock to fall from the tree.
Next, let's find the time it takes for the rock to sink to the bottom of the lake. The speed at which the rock sinks is given as 1.6 m/s.
Using the formula:
t = d / v
Where:
t = time
d = distance
v = velocity/speed
Here, the distance is the depth of the lake, which is 4.6 m, and the speed is given as 1.6 m/s. Plugging in the values, we have:
t = 4.6 m / 1.6 m/s
t ≈ 2.875 s
So, it takes approximately 2.875 seconds for the rock to sink to the bottom of the lake.
To find the total elapsed time, we add the time it takes for the rock to fall from the tree to the time it takes for the rock to sink to the bottom of the lake:
Total elapsed time = time to fall + time to sink
Total elapsed time ≈ 2.02 s + 2.875 s
Total elapsed time ≈ 4.895 s
Therefore, the total elapsed time is approximately 4.895 seconds.
h = 0.5g*t1^2 = 20 m.
4.9*t1^2 = 20
t1^2 = 4.08
t1 = 2.02 s.
d = r * t2 = 4.6 m.
1.6t2 = 4.6
t2 = 2.9 s.
T = t1 + t2 = 2.02 + 2.9 = 4.92 s.