Problem 62 - Calculus Single Variable

Question

Visitors to the top of the Empire State Building in New York City are cautioned not to throw objects off the roof. How much damage could a one ounce weight dropped from the roof do? The height of the building is 1250 feet. Using the quantity mass \(\times\) velocity, which is momentum, as a measure of the impact of the weight when it strikes, compare your answer with the impact of a ten ounce hammer being swung at a speed of ten feet per second.

Step-by-Step Solution

Answer

The one ounce weight has more momentum (17.73 lb-ft/s) compared to a 10 ounce hammer at 10 ft/s (6.25 lb-ft/s), indicating greater impact.

Step 1: Convert Units

Before comparing momentum, we need to convert the mass from ounces to pounds, as weight is typically given in pounds in the imperial system for force calculations. Since 1 pound is equal to 16 ounces, the mass of the one ounce weight is \( \frac{1}{16} \) pounds.

Step 2: Calculate Velocity of Falling Object

Next, calculate the velocity of the weight when it hits the ground using the formula for free fall: \( v = \sqrt{2gh} \), where \( g = 32.2 \text{ ft/s}^2 \) is the acceleration due to gravity and \( h = 1250 \text{ ft} \) is the height. So, \[ v = \sqrt{2 \times 32.2 \times 1250} = \sqrt{80500} \approx 283.7 \text{ ft/s}. \]

Step 3: Calculate Momentum of the Falling Object

Momentum is given by \( p = m \times v \). For the one ounce weight, \( m = \frac{1}{16} \text{ lb} \) and \( v = 283.7 \text{ ft/s} \). Therefore, the momentum is \[ p = \frac{1}{16} \times 283.7 = 17.73 \text{ lb-ft/s}. \]

Step 4: Calculate Momentum of the Hammer

For the ten ounce hammer, first convert the weight to pounds: \( \frac{10}{16} = 0.625 \text{ lb} \). The given speed is \( 10 \text{ ft/s} \). Thus, the momentum of the hammer is \[ p = 0.625 \times 10 = 6.25 \text{ lb-ft/s}. \]

Step 5: Compare Momentum Values

The calculated momentum of the falling one ounce weight is \( 17.73 \text{ lb-ft/s} \), while the hammer's momentum when swung is \( 6.25 \text{ lb-ft/s} \). Therefore, the falling weight has significantly higher momentum.

Key Concepts

Unit Conversion

To begin, understanding unit conversion is crucial in physics calculations. In this case, we're dealing with weights measured in ounces. However, when calculating forces or momentum, it's common to use pounds. This is because pounds directly relate to force calculations in the imperial system. When we convert ounces to pounds, it's important to remember this simple conversion factor:

• 1 pound = 16 ounces

For example, if we have a 1-ounce weight, its equivalent in pounds is:\[m = \frac{1}{16} \text{ pounds}\]Knowing how to convert units accurately ensures you start your calculations on the right foot.

Velocity Calculation

Calculating the velocity of a falling object is an essential step in determining its impact when it hits the ground. An object in free fall accelerates downwards due to gravity, which has a specific value:

• Acceleration due to gravity \( g = 32.2 \text{ ft/s}^2 \)

For an object dropped from a height, you can determine its final velocity using the formula:\[ v = \sqrt{2gh} \]Where:

• \( g \) is the acceleration due to gravity
• \( h \) is the height from which the object is dropped, \( h = 1250 \text{ ft} \)

Plugging in these values, we calculate:\[ v = \sqrt{2 \times 32.2 \times 1250} = 283.7 \text{ ft/s} \]This shows us how quickly the object is moving when it hits the ground, which is necessary for further calculations.

Free Fall

The concept of free fall relates to how gravity affects an object in motion, with no other forces acting on it except for this force of gravity. In this exercise, we deal with the idea of a weight dropped from a height. A free-falling object has a straightforward motion, as its velocity continuously increases while falling until it strikes the ground. This simplified physics scenario helps you predict outcomes using the basic physics formulae, such as the velocity calculation mentioned before. Remember, in real-world physics, factors like air resistance can affect motion. However, in typical exercises, we often consider an "idealized" scenario, assuming no air resistance to focus solely on gravitational effects.

Momentum Comparison

Momentum gives us a measure of the effect or damage an object could impart when it hits another object. It is calculated using:\[ p = m \times v \]Where:

• \( m \) is the mass of the object
• \( v \) is its velocity

For the weight, its calculated momentum was \( 17.73 \text{ lb-ft/s} \). In comparison, the hammer's momentum was much less, \( 6.25 \text{ lb-ft/s} \), even though it was much lighter. This comparison shows how velocity affects momentum considerably more. A small object can have greater impact due entirely to its fast speed rather than its mass. Understanding momentum allows students to grasp why sometimes small fast-moving objects can cause as much or more damage than slower yet heavier objects.