Problem 43 - Prealgebra

Question

PHYSICS The formula \(h=16 t^{2}\) measures the time \(t\) in seconds that it takes for an object to fall from a height of \(h\) feet and hit the ground. How long would it take a marble to hit the ground if it was dropped off a cliff with a height of 150 feet? Round to the nearest tenth.

Step-by-Step Solution

Answer

It would take approximately 3.1 seconds for the marble to hit the ground.

Step 1: Identify the Known Values

The problem provides the height \(h = 150\) feet. We're tasked with finding the time \(t\) when the marble reaches the ground.

Step 2: Apply the Formula

Use the given formula \(h = 16t^2\). Substitute \(h = 150\) into the equation, so it becomes \(150 = 16t^2\).

Step 3: Solve for \(t^2\)

Rearrange the formula to solve for \(t^2\) by dividing both sides by 16: \(t^2 = \frac{150}{16}\).

Step 4: Calculate \(t^2\) Value

Perform the division: \(t^2 = 9.375\).

Step 5: Find \(t\) by Taking the Square Root

To solve for \(t\), take the square root of both sides: \(t = \sqrt{9.375}\).

Step 6: Calculate and Round the Time

Calculate \(t = \sqrt{9.375} \approx 3.06\). Round 3.06 to the nearest tenth to get \(t = 3.1\).

Key Concepts

Gravity

Gravity is a force that attracts two bodies towards each other. The most common experience we have with gravity is the pull that the Earth has on objects, pulling them towards the ground. In the context of physics problems, especially those involving free fall, gravity is typically treated as a constant force.

• The acceleration due to gravity on Earth is approximately 32 feet per second squared (ft/s²), or about 9.8 meters per second squared (m/s²) in different units.
• This means that in the absence of other forces, an object will speed up by this rate every second as it falls.

Understanding gravity is crucial when solving physics problems that involve motion, like the one in our exercise. It helps explain why objects fall to the ground when dropped and sets the stage for calculating the time taken to reach the ground.

Free Fall

Free fall refers to the motion of an object when it is only acted upon by gravity. In this situation, other forces, like air resistance, are neglected to simplify calculations and focus on gravity's effects.

• When an object is in free fall, it will accelerate downwards at a rate equal to the acceleration due to gravity.
• The formula used in the exercise, \( h = 16t^2 \), assumes that the only force acting on the marble is gravity.
• This formula was derived under the assumption of free fall, meaning air resistance and other forces are ignored.

Understanding the concept of free fall is essential because it helps us calculate the time it takes for an object to fall a certain height. By recognizing that the formula comes from the laws of motion under gravity, students can solve similar physics problems with confidence.

Problem Solving

When approaching any physics problem, especially those involving motion, having a structured method is crucial. Let's use the marble falling exercise as a guide to understand problem solving in physics.
Firstly, identify known values. In the exercise, you are given the height from which the marble falls. This initial step helps clarify what you know and what you need to find.

• Then, use the applicable formula. Recognizing that \( h = 16t^2 \) relates height and time under free fall is key.
• Substitute the known values, and rearrange the equation to solve for the unknown, which in this case is time \( t \).
• Simplify and calculate. By breaking down each part of the formula application, students can check their work for errors and ensure correct calculations.
• Finally, ensure your final answer is sensible, both in magnitude and units, rounding as necessary based on the context, such as the step where the time was rounded to the nearest tenth.

Being methodical and organized when solving physics problems allows you to tackle complex scenarios systematically and with greater confidence.