Let's dive into the physics of calculating the work needed to launch a 6 oz object to a speed of 160 ft/sec. This is a classic problem involving the work-energy principle, which is a fundamental concept in mechanics. Understanding this principle not only helps in solving physics problems but also provides insights into how energy is transferred and transformed in various real-world scenarios. We'll break down the problem step by step, ensuring that you grasp each concept along the way. So, if you're ready to explore the fascinating world of energy and motion, let's get started!
Understanding the Work-Energy Principle
The work-energy principle is a cornerstone of physics, linking the work done on an object to its change in kinetic energy. In simpler terms, it states that the net work done on an object is equal to the change in its kinetic energy. This principle is incredibly useful because it allows us to relate forces and motion without directly dealing with time, making many problems significantly easier to solve. When we talk about work in physics, we're referring to the energy transferred to or from an object by a force acting on it. This transfer of energy results in a change in the object's state of motion. The formula that encapsulates this principle is:
W = ΔKE
Where:
Wis the work done on the object.ΔKEis the change in kinetic energy.
Kinetic energy, on the other hand, is the energy an object possesses due to its motion. It depends on both the mass of the object and its velocity. The formula for kinetic energy (KE) is:
KE = (1/2) * m * v^2
Where:
mis the mass of the object.vis the velocity of the object.
To effectively apply the work-energy principle, it's crucial to understand these concepts thoroughly. This principle bridges the gap between force, motion, and energy, allowing us to analyze a wide range of physical phenomena, from a simple push on a box to the complex dynamics of a rocket launch. In our problem, we'll use this principle to determine the work required to accelerate the 6 oz object to 160 ft/sec.
Problem Setup and Mass Conversion
Before we jump into calculations, let's set up the problem and ensure we have all the necessary information in the correct units. We are given that the object has a weight of 6 oz and is initially at rest. Our goal is to find the work required to accelerate this object to a final velocity of 160 ft/sec. The first crucial step is to convert the weight from ounces to a unit of mass that is consistent with the units of velocity (ft/sec) and work (ft-lb). In the English system of units, the appropriate unit for mass is the slug. We know that weight (W) is related to mass (m) and the acceleration due to gravity (g) by the formula:
W = m * g
Where:
Wis the weight in pounds (lb).mis the mass in slugs.gis the acceleration due to gravity, approximately 32.2 ft/sec².
However, our weight is given in ounces, so we need to convert it to pounds first. There are 16 ounces in a pound, so:
Weight (in pounds) = 6 oz / 16 oz/lb = 0.375 lb
Now we can use the formula W = m * g to find the mass in slugs:
0. 375 lb = m * 32.2 ft/sec²
Solving for m:
m = 0.375 lb / 32.2 ft/sec² ≈ 0.0116 slugs
So, the mass of the object is approximately 0.0116 slugs. This conversion is essential because it ensures that our calculations are dimensionally consistent, and we'll obtain the correct units for work (ft-lb). With the mass in slugs, we can now proceed to calculate the kinetic energy and, subsequently, the work required.
Calculating Kinetic Energy
Now that we have the mass of the object in slugs (0.0116 slugs) and the final velocity (160 ft/sec), we can calculate the final kinetic energy. Remember, kinetic energy (KE) is the energy an object possesses due to its motion, and it's given by the formula:
KE = (1/2) * m * v^2
Plugging in the values we have:
KE = (1/2) * 0.0116 slugs * (160 ft/sec)^2
Let's break down the calculation:
KE = (1/2) * 0.0116 * 25600
KE = 0.0058 * 25600
KE ≈ 148.48 ft-lb
So, the final kinetic energy of the object is approximately 148.48 ft-lb. But remember, we need to find the change in kinetic energy. Since the object was initially at rest, its initial kinetic energy was zero. Therefore, the change in kinetic energy (ΔKE) is simply the final kinetic energy minus the initial kinetic energy (which is zero):
ΔKE = KE_final - KE_initial
ΔKE = 148.48 ft-lb - 0 ft-lb
ΔKE = 148.48 ft-lb
This result is crucial because, according to the work-energy principle, the work done on the object is equal to this change in kinetic energy. In the next section, we'll connect this result to the work required to project the object.
Applying the Work-Energy Principle
We've calculated the change in kinetic energy (ΔKE) to be approximately 148.48 ft-lb. Now, we apply the work-energy principle, which, as we discussed earlier, states that the work done on an object is equal to its change in kinetic energy:
W = ΔKE
In our case:
W = 148.48 ft-lb
This means the work required to project the 6 oz object from rest to a velocity of 160 ft/sec is 148.48 ft-lb. However, the question asks us to round the final answer to the nearest tenth. So, let's round 148.48 ft-lb to the nearest tenth:
W ≈ 148.5 ft-lb
Therefore, the work required is approximately 148.5 ft-lb. This final step of rounding ensures we provide the answer in the format requested by the problem. It's always important to pay attention to the specific instructions regarding rounding and units to ensure your answer is complete and accurate.
Final Answer
After meticulously working through the problem, from converting units to applying the work-energy principle and rounding our final answer, we've arrived at the solution. The work required to project a 6 oz object initially at rest to 160 ft/sec is approximately:
148. 5 ft-lb
This answer represents the amount of energy that needs to be transferred to the object to achieve the desired velocity. By understanding the work-energy principle and carefully performing each step of the calculation, we've successfully solved this physics problem. Remember, physics is all about understanding the underlying principles and applying them methodically. So, keep practicing and exploring, and you'll master these concepts in no time!
In summary, we first converted the weight of the object from ounces to slugs to ensure consistent units. Then, we calculated the final kinetic energy using the formula KE = (1/2) * m * v^2. Applying the work-energy principle, we equated the work done to the change in kinetic energy. Finally, we rounded the answer to the nearest tenth as requested. This step-by-step approach not only helps in solving problems but also deepens your understanding of the physics concepts involved.