Problem 28 - Calculus Single Variable

Question

An object is dropped from a window \(100 \mathrm{ft}\) above the ground. At what speed is the object traveling at the moment of impact with the ground?

Step-by-Step Solution

Answer

The object's speed at impact is 80 ft/s.

Step 1: Understand the Problem

We need to find the speed of an object just before it hits the ground when it is dropped from a height of 100 feet.

Step 2: Use the Formula for Speed at Impact

To determine the speed at impact, we'll use the formula for an object under free fall: \[ v = \sqrt{2gh} \]where:- \( g \) is the acceleration due to gravity (approximately \( 32 \, \text{ft/s}^2 \)),- \( h \) is the height from which the object is dropped (100 ft in this case).

Step 3: Plug in the Values

Substitute \( g = 32 \, \text{ft/s}^2 \) and \( h = 100 \, \text{ft} \) into the formula:\[ v = \sqrt{2 \times 32 \times 100} \]

Step 4: Calculate the Expression

Compute the expression inside the square root first:\[ 2 \times 32 \times 100 = 6400 \]Then take the square root:\[ v = \sqrt{6400} = 80 \, \text{ft/s} \]

Key Concepts

Acceleration Due to Gravity

When we talk about objects in free fall, acceleration due to gravity is a crucial factor. This is a force that pulls objects toward the Earth's center. In physics, this acceleration is denoted by the symbol \( g \).
Free falling objects near Earth accelerating due to gravity have a constant \( g \) value of about \( 32 \, \text{ft/s}^2 \). This means that for every second an object is in free fall, its speed increases by approximately 32 feet per second.

• It acts on all objects, regardless of their mass.
• It guides the object's motion from rest to higher speeds.
• The direction of this acceleration is always downward toward Earth's center.

To emphasize, gravity is what gives weight to objects and causes the phenomenon of free fall. Remembering this will help understand how freely falling objects gain speed over time.

Velocity Calculation

Velocity in the context of free fall refers to the speed of an object along with its direction just before it strikes the ground. When an object is released from rest, as in our given problem, we can calculate this velocity using specific formulas.
In physics, the formula to calculate the velocity of a falling object just before impact is: \[ v = \sqrt{2gh} \]Here, \( v \) is the velocity, \( g \) is the acceleration due to gravity, and \( h \) is the height from which the object is dropped. The square root signifies that we derive the speed based on distance and acceleration. This approach specifically relates to free-falling objects, where air resistance is negligible.

• The velocity equation ties together height and gravity directly.
• It forms the basis for determining how fast an object will be traveling just before impact.
• Using these variables accurately ensures correct results.

Staying mindful of units and properly applying the formula will allow you to compute velocity readings systematically.

Impact Speed

Impact speed refers to the speed at which an object strikes the ground after falling. This is the final velocity just before the object hits a surface. It is critical in understanding how fast an object was traveling when it meets the ground.
For our problem, given that the object falls from 100 feet, the impact speed was calculated as 80 \( \text{ft/s} \) using the velocity formula outlined earlier. This value signifies how rapidly the object was traveling right before the collision with the ground.

• Impact speed helps predict the severity of a collision.
• It provides insights into energy exerted during impact.
• The greater the height from which an object falls, the higher the impact speed, given constant gravitational acceleration.

Knowing the impact speed is not only a matter of calculating numbers; it has real-world implications in fields such as safety engineering and materials science. Each time an object impacts at high speed, energy is transferred, causing potential damage or change of momentum.