An Air Pump Performs 5,600 J Of Work To Launch A Water Bottle Rocket Into The Air, Applying 150 N Of Force At A 45-degree Angle. What Is The Horizontal Distance The Rocket Travels?

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The fascinating world of physics comes alive when we explore seemingly simple activities like launching a water bottle rocket. These rockets, propelled by compressed air, demonstrate fundamental physics principles such as work, force, and displacement. This article delves into a scenario where an air pump performs 5,600 Joules (J) of work to launch a water bottle rocket. Given a force of 150 Newtons (N) applied at a 45-degree angle to the ground, we aim to calculate the horizontal distance the rocket travels. This exploration will not only reveal the mechanics behind the launch but also highlight the interconnectedness of various physics concepts.

Understanding Work, Force, and Displacement

To accurately calculate the horizontal distance, we must first establish a strong understanding of the core concepts at play: work, force, and displacement. These concepts form the bedrock of mechanics and are essential for analyzing the motion of objects, including our water bottle rocket.

Work, in physics, is defined as the energy transferred to or from an object by the application of a force along with a displacement. It's a scalar quantity, meaning it has magnitude but no direction, and is measured in Joules (J). The work done on an object is directly related to the force applied and the distance over which the force acts. Mathematically, work (W) is expressed as:

W = F ⋅ d ⋅ cos(θ)

Where:

  • W represents work done (in Joules)
  • F represents the magnitude of the force (in Newtons)
  • d represents the magnitude of the displacement (in meters)
  • θ represents the angle between the force vector and the displacement vector

This equation highlights a crucial aspect of work: only the component of force acting in the direction of displacement contributes to the work done. When the force and displacement are in the same direction (θ = 0°), the work done is maximized. If the force is perpendicular to the displacement (θ = 90°), no work is done.

Force, on the other hand, is a vector quantity that describes an interaction that, when unopposed, will change the motion of an object. It has both magnitude and direction and is measured in Newtons (N). A force can cause an object to accelerate, decelerate, or change direction. In the context of our water bottle rocket, the force applied by the air pump is what propels the rocket forward. This force overcomes inertia and imparts motion to the rocket.

The concept of force is intrinsically linked to Newton's Laws of Motion. The first law states that an object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction unless acted upon by a force. The second law quantifies the relationship between force, mass, and acceleration: F = ma, where 'm' is mass and 'a' is acceleration. The third law states that for every action, there is an equal and opposite reaction. Understanding these laws provides a framework for analyzing how forces influence the motion of objects.

Displacement is the vector quantity that refers to the change in position of an object. It is defined as the shortest distance between the initial and final positions of the object and has both magnitude and direction. Displacement is different from distance, which is the total length of the path traveled. In our scenario, we are particularly interested in the horizontal displacement of the water bottle rocket, which is the horizontal distance it covers from its launch point to its landing point.

Displacement is a crucial concept in understanding motion because it allows us to describe how an object's position changes over time. Along with velocity and acceleration, displacement forms the cornerstone of kinematics, the study of motion without considering its causes. Understanding displacement is vital for accurately predicting and analyzing the trajectory of the water bottle rocket.

By grasping these fundamental concepts – work, force, and displacement – we lay the groundwork for a deeper analysis of the water bottle rocket launch. We can now proceed to apply these principles to calculate the horizontal distance traveled by the rocket, unraveling the physics behind its flight.

Applying the Work-Energy Theorem

The work-energy theorem provides a powerful link between the work done on an object and its change in kinetic energy. This theorem is a direct consequence of Newton's second law of motion and offers an alternative approach to analyzing motion, especially when forces are not constant or the motion is complex. It states that the net work done on an object is equal to the change in its kinetic energy. Mathematically, this can be expressed as:

W_net = ΔKE = KE_f - KE_i

Where:

  • W_net is the net work done on the object
  • ΔKE is the change in kinetic energy
  • KE_f is the final kinetic energy
  • KE_i is the initial kinetic energy

Kinetic energy (KE) is the energy possessed by an object due to its motion and is given by the formula:

KE = (1/2) * m * v^2

Where:

  • m is the mass of the object
  • v is the velocity of the object

In the context of our water bottle rocket launch, the work done by the air pump (5,600 J) is converted into the kinetic energy of the rocket. Assuming the rocket starts from rest (KE_i = 0), the work-energy theorem tells us that the work done is equal to the final kinetic energy of the rocket. This allows us to determine the rocket's velocity immediately after the air pump has done its work.

However, we must carefully consider the angle at which the force is applied. The 150 N force is applied at a 45-degree angle to the ground. This means that only a component of the force is contributing to the horizontal displacement. To accurately apply the work-energy theorem, we need to relate the work done to the horizontal component of the displacement.

The work done can be expressed as the product of the force component in the direction of displacement and the displacement itself. In this case, the horizontal component of the force is Fcos(θ), where θ is the angle between the force and the horizontal direction. Therefore, the work done can also be written as:

W = F * cos(θ) * d

Where d is the magnitude of the displacement in the direction of the force. However, this d is not the horizontal distance we are trying to find. It's the distance over which the force is applied, which is related to the length of the pump's stroke. We need to connect this to the horizontal distance the rocket travels after it's launched.

Therefore, to effectively use the work-energy theorem for our problem, we will first determine the initial velocity of the rocket after the air pump has completed its work. Then, we will employ projectile motion principles to calculate the horizontal range, bridging the gap between the work done and the final distance traveled by the rocket.

Calculating the Initial Velocity

To determine the horizontal distance the water bottle rocket travels, a crucial first step is calculating the initial velocity imparted to the rocket by the air pump. This velocity is the launching point for understanding the rocket's trajectory and range. We will leverage the work-energy theorem, as discussed earlier, to connect the work done by the air pump to the rocket's kinetic energy.

Recall that the work-energy theorem states that the net work done on an object equals its change in kinetic energy (W_net = ΔKE). In our scenario, the air pump does 5,600 J of work on the rocket. Assuming the rocket starts from rest, its initial kinetic energy (KE_i) is zero. Therefore, the work done by the pump directly translates to the rocket's final kinetic energy (KE_f).

We can express this as:

W = KE_f = (1/2) * m * v^2

Where:

  • W = 5,600 J (work done by the air pump)
  • m = mass of the water bottle rocket (This information is missing from the problem statement and needs to be assumed or measured experimentally. For the sake of demonstration, let's assume the mass of the rocket, m, is 0.5 kg)
  • v = the initial velocity of the rocket (what we want to find)

Substituting the values, we get:

5,600 J = (1/2) * 0.5 kg * v^2

Solving for v:

v^2 = (5,600 J * 2) / 0.5 kg
v^2 = 22,400 m^2/s^2
v = √(22,400 m^2/s^2)
v ≈ 149.67 m/s

Therefore, the initial velocity (v) of the water bottle rocket as it leaves the air pump is approximately 149.67 meters per second. This is the magnitude of the initial velocity vector. However, remember that the force was applied at a 45-degree angle. This means the initial velocity has both horizontal and vertical components.

To analyze the rocket's trajectory using projectile motion principles, we need to decompose this initial velocity into its horizontal (v_x) and vertical (v_y) components. Since the launch angle is 45 degrees, these components are equal:

v_x = v * cos(45°)
v_y = v * sin(45°)

Since cos(45°) = sin(45°) ≈ 0.707:

v_x ≈ 149.67 m/s * 0.707 ≈ 105.79 m/s
v_y ≈ 149.67 m/s * 0.707 ≈ 105.79 m/s

Thus, the initial horizontal velocity component (v_x) is approximately 105.79 m/s, and the initial vertical velocity component (v_y) is also approximately 105.79 m/s. These values are crucial for the next step: calculating the horizontal range of the water bottle rocket.

Calculating the Horizontal Distance (Range)

Now that we have determined the initial velocity components of the water bottle rocket, we can calculate the horizontal distance it travels, also known as the range. This calculation involves applying principles of projectile motion, which describes the curved path an object follows when launched into the air.

Projectile motion analysis typically assumes that the only force acting on the object after launch is gravity (neglecting air resistance). This simplifies the problem, allowing us to use well-established kinematic equations. The horizontal motion and vertical motion are treated independently, as gravity only affects the vertical motion.

The horizontal distance (range, R) of a projectile launched with an initial velocity v at an angle θ can be calculated using the following formula:

R = (v^2 * sin(2θ)) / g

Where:

  • R is the range (horizontal distance)
  • v is the initial velocity
  • θ is the launch angle
  • g is the acceleration due to gravity (approximately 9.8 m/s²)

In our case, we have:

  • v ≈ 149.67 m/s (initial velocity)
  • θ = 45° (launch angle)
  • g = 9.8 m/s²

Substituting these values into the formula:

R = ((149.67 m/s)^2 * sin(2 * 45°)) / 9.8 m/s²
R = ((149.67 m/s)^2 * sin(90°)) / 9.8 m/s²

Since sin(90°) = 1:

R = (22,401.11 m²/s²) / 9.8 m/s²
R ≈ 2285.83 m

Therefore, the horizontal distance (range) the water bottle rocket travels is approximately 2285.83 meters. This is a significant distance, highlighting the impressive energy transfer from the air pump to the rocket's motion.

It's important to note that this calculation assumes ideal conditions, neglecting factors like air resistance. In a real-world scenario, air resistance would play a significant role, slowing the rocket down and reducing its range. The actual distance traveled would likely be less than our calculated value.

This result demonstrates how physics principles can be applied to analyze and predict the behavior of objects in motion. By understanding the concepts of work, energy, and projectile motion, we can gain insights into the mechanics of everyday phenomena, such as launching a water bottle rocket.

In conclusion, the journey of calculating the horizontal distance traveled by a water bottle rocket launched with 5,600 J of work and a 150 N force at a 45-degree angle showcases the interconnectedness of fundamental physics principles. We successfully determined the horizontal distance to be approximately 2285.83 meters by meticulously applying the concepts of work, energy, and projectile motion.

We began by defining key terms such as work, force, and displacement, establishing the foundation for our analysis. We then explored the work-energy theorem, which allowed us to relate the work done by the air pump to the rocket's initial kinetic energy. By calculating the initial velocity and its horizontal and vertical components, we set the stage for projectile motion analysis. Finally, using the projectile motion equations, we determined the horizontal range, the ultimate goal of our investigation.

This exercise not only provides a numerical answer but also emphasizes the importance of a systematic approach to problem-solving in physics. Breaking down a complex problem into smaller, manageable steps, understanding the underlying principles, and applying appropriate equations are crucial skills in physics and other scientific disciplines.

Furthermore, this analysis highlights the role of assumptions and simplifications in physics modeling. We neglected air resistance to simplify the calculations, but in reality, air resistance would significantly affect the rocket's trajectory and range. This underscores the importance of considering the limitations of our models and the potential impact of real-world factors.

The exploration of water bottle rocket launches serves as an engaging way to learn and appreciate physics. It demonstrates how abstract concepts can be applied to tangible situations, making learning more meaningful and memorable. By understanding the physics behind such activities, we gain a deeper understanding of the world around us and develop critical thinking skills that are valuable in various aspects of life.

This analysis can be extended by considering factors such as air resistance, the effect of wind, and the efficiency of the air pump. These extensions would provide a more realistic and comprehensive understanding of the water bottle rocket launch and further solidify the application of physics principles.

In essence, calculating the horizontal distance of a water bottle rocket launch is more than just a physics problem; it's an exploration of fundamental principles, a demonstration of problem-solving techniques, and a testament to the power of physics in explaining the world we experience.