Question

A rock dropped from a 1024-foot-high cliff falls a distance \(D\) given by \(D=16 t^2\), where \(t\) is the time in seconds after the rock is dropped. How long will it take the rock to reach the bottom of the cliff? A. 64 seconds B. 32 seconds C. 16 seconds D. 8 seconds

Short answer

D. 8 seconds

Step-by-step solution

Identify the given information

The cliff is 1024 feet high. The distance the rock falls, \(D\), is given by the equation \(D = 16t^2\).

Set up the equation

Since the rock is falling from a height of 1024 feet, we set \(D\) to 1024. So, the equation becomes \(1024 = 16t^2\).

Solve for \(t^2\)

Divide both sides of the equation by 16 to isolate \(t^2\): \(\frac{1024}{16} = t^2\), which simplifies to \(64 = t^2\).

Take the square root

Take the square root of both sides to solve for \(t\): \(t = \sqrt{64}\). This results in \(t = 8\).

Verify the correct answer choice

The time it takes for the rock to reach the bottom of the cliff is 8 seconds. Therefore, the correct answer is D. 8 seconds.

Key concepts

physics problems

Physics problems can often be a bit tricky but exciting! They help us understand the laws of nature. In this exercise, we are dealing with the motion of a rock falling from a cliff. This type of motion is called 'free fall' and is heavily influenced by gravity. When an object is in free fall, it accelerates downwards at a constant rate due to gravity. The formula used here, \(D = 16t^2\), combines this acceleration and the time to tell us how far the rock will travel in a given time. The number 16 here actually comes from the simplification of the gravitational acceleration on Earth (\textrm{32 ft/s²} divided by 2). This makes it easier to predict the distance an object will fall.

distance and time equations

Equations that relate distance and time are fundamental in both math and physics. They allow us to calculate one quantity if we know the other. In our case, the equation \(D = 16t^2\) relates the distance \(D\) fallen by the rock to the time that has passed since it was dropped. To solve the problem, we substituted the known distance (1024 feet) into the equation and solved for time (\(t\)). Here’s how it worked:

• Identify the given values and equation: 1024 feet and \(D = 16t^2\).
• Substitute the distance: \(1024 = 16t^2\).
• Isolate \(t^2\): \(\frac{1024}{16} = t^2\), which simplifies to \(64 = t^2\).

This leaves you with a simple last step involving square roots.

square root calculation

Taking square roots is a necessary skill in many areas of math and science. It helps us find the original value before it was squared. To isolate the time \(t\) in our problem, we used the square root. Here's how:

• We had the equation \(64 = t^2\).
• Taking the square root of both sides, we get \(t = \sqrt{64}\).
• Simplifying \sqrt{64}, we find \(t = 8\).

Always remember, the square root of a number is a value that, when multiplied by itself, gives the original number. In our case, \(\frac{64}{8} = 8\), confirming that \(t = 8\) seconds is indeed correct. This final step is crucial because it converts mathematical answers into real-world time units, clarifying that the rock takes 8 seconds to hit the ground.