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Chapter 9: Problem 16

A 10 -m-Iong glider with a mass of \(680 \mathrm{kg}\) (including the passengers) is gliding horizontally through the air at \(30 \mathrm{m} / \mathrm{s}\) when a \(60 \mathrm{kg}\) skydiver drops out by releasing his grip on the glider. What is the glider's velocity just after the skydiver lets go?

Short Answer

Expert verified
The velocity of the glider just after the skydiver lets go can be found using the formula \(V_{final} = \frac{M_{initial} \times V_{initial}}{M_{final}}\). From this we calculate \(V_{final}\) as the correct answer to the problem.

Step by step solution

01

Identify the initial state of the system

Before the skydiver jumps off, the total mass of the system is the mass of the glider and the skydiver together. The initial momentum of the system is given by the product of the total mass and the initial velocity of the glider. So, calculate the total initial mass (\(M_{initial}\)) as the sum of the mass of the glider (\(M_{glider}\)) and the mass of the skydiver (\(M_{skydiver}\)). The initial velocity is given as \(V_{initial}\). Hence, the initial momentum \(P_{initial}\) can be calculated as \(P_{initial} = M_{initial} \times V_{initial}\)
02

Identify the final state of the system

After the skydiver jumps off, the total mass of the system is just the mass of the glider and its velocity is what we need to find. So, the final total mass (\(M_{final}\)) is the mass of the glider. The final momentum \(P_{final}\), according to the principle of conservation of linear momentum, should be equal to \(P_{initial}\).
03

Apply the principle of conservation of linear momentum

Set the initial and final momentum equal to each other and solve for the final velocity of the glider (\(V_{final}\)). The equation will be \(P_{initial} = P_{final}\), which turns into \(M_{initial} \times V_{initial} = M_{final} \times V_{final}\).
04

Solve for the final velocity

Solving the above equation, the final velocity (\(V_{final}\)) of the glider can be found by dividing both sides of the equation by \(M_{final}\) giving us the equation \(V_{final} = \frac{M_{initial} \times V_{initial}}{M_{final}}\). The values of \(M_{initial}\), \(V_{initial}\) and \(M_{final}\) are known to us, which can be substituted into the equation to find \(V_{final}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Momentum in Physics
Momentum, in physics, is essentially the quantity of motion possessed by an object. It is a vector quantity, meaning it has both magnitude and direction. The linear momentum of an object is calculated by the product of its mass (in kilograms) and its velocity (in meters per second), represented by the formula
\( p = m \times v \).
For example, if a car with a mass of 1000kg is moving with a velocity of 20m/s, its momentum would be 20000kg\(\cdot\)m/s. Momentum is crucial in analyzing situations where collisions or separations, like that of a skydiver leaving a glider, occur.
Glider Velocity
The velocity of a glider refers to how fast it's moving along a straight path and in which direction. Measured in meters per second (m/s), glider velocity plays a key role when calculating the momentum of a system. When a skydiver leaves the glider, we expect changes in velocity due to the conservation of momentum. The velocity just after the skydiver departs is crucial for determining the new state of motion for the glider.
Principle of Conservation of Momentum
The Principle of Conservation of Momentum states that the total linear momentum of an isolated system remains constant provided no external forces are acting on it. This principle helps us analyze circumstances where two or more objects interact. In the absence of external forces, the initial momentum, which is the combined momentum of the glider and skydiver before separation, will be equal to the final momentum of the glider right after the skydiver's exit. Formulated as \( P_{initial} = P_{final} \), this principle allows us to solve for unknown quantities, such as the final velocity, given the mass and velocity before the event.
System Mass Calculation
Calculating system mass is simply finding the total mass of all the components within a system. It involves adding up the individual masses of these components. When a skydiver is holding onto a glider, both their masses contribute to the system's mass. For example, with a glider mass of 680kg and a skydiver mass of 60kg, the total system mass will be 740kg, which impacts the initial momentum calculation. After the skydiver exits, the mass of the system reduces, and it's just the mass of the glider that must be considered for further calculations. Understanding this is essential for correctly applying the principle of conservation of momentum to find new velocities after a mass change.

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Most popular questions from this chapter

The Army of the Nation of Whynot has a plan to propel small vehicles across the battlefield by shooting them, from behind, with a machine gun. They've hired you as a consultant to help with an upcoming test. A \(100 \mathrm{kg}\) test vehicle will roll along frictionless rails. The vehicle has a tall "sail" made of ultrahard steel. Previous tests have shown that a \(20 \mathrm{g}\) bullet traveling at \(400 \mathrm{m} / \mathrm{s}\) rebounds from the sail at \(200 \mathrm{m} / \mathrm{s}\). The design objective is for the cart to reach a speed of \(12 \mathrm{m} / \mathrm{s}\) in \(20 \mathrm{s}\). You need to tell them how many bullets to fire per second. A five-star general will watch the test, so your ability to get future consulting jobs will depend on the outcome.

What is the magnitude of the momentum of a. A \(1500 \mathrm{kg}\) car traveling at \(10 \mathrm{m} / \mathrm{s} ?\) b. \(A 200 g\) baseball thrown at \(40 \mathrm{m} / \mathrm{s} ?\)

Two ice skaters, with masses of \(50 \mathrm{kg}\) and \(75 \mathrm{kg}\), are at the center of a \(60-\) m-diameter circular rink. The skaters push off against each other and glide to opposite edges of the rink. If the heavier skater reaches the edge in \(20 \mathrm{s}\), how long does the lighter skater take to reach the edge?

A two-stage rocket is traveling at \(1200 \mathrm{m} / \mathrm{s}\) with respect to the earth when the first stage runs out of fuel. Explosive bolts release the first stage and push it backward with a speed of \(35 \mathrm{m} / \mathrm{s}\) relative to the second stage. The first stage is three times as massive as the second stage. What is the speed of the second stage after the separation?

A 20 g ball of clay is shot to the right (in the positive \(x\) direction) at \(12 \mathrm{m} / \mathrm{s}\) toward a \(40 \mathrm{g}\) ball of clay at rest. The two balls of clay collide and stick together. Call this reference frame \(S\). a. What is the total momentum in frame S? b. What is the velocity \(\vec{V}\) of a reference frame \(S^{\prime}\) in which the total momentum is zero? c. After the collision, what is the velocity in frame S' of the resulting \(60 \mathrm{g}\) ball of clay? Answering this question requires only thought, no calculations. d. Use your answer to part \(c\) and the Galilean transformation of velocity to find the post-collision velocity of the \(60 \mathrm{g}\) ball of clay in reference frame S.
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