The Alert Hiker: An alert hiker sees a boulder fall from the top of a distant cliff and notes that it takes 1.3s for the boulder to fall the last third of the way to the ground.
a.What is the height of the cliff in meters?
b.If in part (a) you get two solutions of a quadratic equation and you use one for your answer, what does the other represent?
Let H be the height of the cliff. The the time required to fall a distance d is
t = sqrt(2d/g)
From the information given you can write:
sqrt(2H/g) - sqrt[(2/3)*2H/g] = 1.3
sqrt(2H/g)*[1 - sqrt(2/3)] = 1.3
0.1835*sqrt(2H/g)= 1.3
sqrt(2H/g) = 7.08
H = 246 m
Check: 2/3 of the way down is 164 m
Time to fall 164 m = 5.785 s
Time to fall 246 m = 7.085 s
Difference = 1.30 s
I did not have to solve a quadratic equation.
To find the height of the cliff, we can use the formulas and information given in the problem. Let's break down the steps:
a. Find the time it takes for the boulder to fall the first two-thirds of the way to the ground:
Since it takes 1.3 seconds for the boulder to fall the last third of the distance, it means it took (1.3 x 3) = 3.9 seconds for the boulder to fall the entire distance.
Let's denote the total time taken as 't'.
Now, we can use the formula for the height dropped by an object under free fall:
h = (1/2)gt²
Since we have the value of 't' (3.9 seconds) and we want to find 'h', we can rearrange the formula as follows:
h = (1/2)gt²
h = (1/2)(9.8 m/s²)(3.9 s)²
h = (1/2)(9.8 m/s²)(15.21 s²)
h ≈ 74.89 meters
Therefore, the height of the cliff is approximately 74.89 meters.
b. If you obtain two solutions for the quadratic equation, it means there are two possible heights for the cliff. One of the solutions represents the actual height of the cliff, while the other solution might be extraneous or incompatible with the given scenario.
To determine which solution is correct, you can consider physical constraints and reasonability. For example, eliminate solutions that are negative or unrealistic in the context of the problem. Remember to consider the given information and constraints to choose the appropriate solution.
To solve this problem, we can use the equations of motion for an object in free fall. We will assume that the boulder falls from rest, without any initial speed.
Let's break down the problem into two parts:
a. To determine the height of the cliff, we can analyze the motion of the boulder during the last third of its fall. We know that it takes 1.3 seconds for the boulder to fall this distance. We'll use the equation for the distance traveled by an object in free fall:
d = (1/2) * g * t^2
Where:
- d is the distance traveled
- g is the acceleration due to gravity, approximately 9.8 m/s^2
- t is the time
We'll denote the height of the cliff as H. Since the boulder falls the last third of the way, we can write the following equation:
(1/3) * H = (1/2) * g * (1.3)^2
Simplifying this equation gives us:
H = (2/3) * g * (1.3)^2
Plugging in the value for g and solving for H will give us the height of the cliff in meters.
b. In some cases, when solving the quadratic equation for the height of the cliff, you may obtain two solutions. One of these solutions will represent the height of the cliff, while the other represents the time it takes for the boulder to reach the top of the cliff if thrown vertically upwards from the same location.
Since the problem doesn't specify whether the boulder was thrown upwards, we can discard the extraneous solution and focus on the height of the cliff.
In summary:
a. Calculate the height of the cliff using H = (2/3) * g * (1.3)^2.
b. Discard any extraneous solution obtained while solving the quadratic equation.