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Chapter 7: Problem 33

A rocket sled with a mass of \(2900 \mathrm{~kg}\) moves at \(250 \mathrm{~m} / \mathrm{s}\) on a set of rails. At a certain point, a scoop on the sled dips into a trough of water located between the tracks and scoops water into an empty tank on the sled. By applying the principle of conservation of translational momentum, determine the speed of the sled after \(920 \mathrm{~kg}\) of water has been scooped up. Ignore any retarding force on the scoop.

Short Answer

Expert verified
Approx. 189.53 m/s

Step by step solution

01

Identify the given values

The given values are: initial mass of the sled, including tank, is 2900 kg; the initial speed of the sled is 250 m/s; and the mass of the water scooped up is 920 kg.
02

Calculate the final total mass

The final mass of the sled includes the initial mass of the sled plus the mass of the water scooped up. \[ m_f = 2900 \text{ kg} + 920 \text{ kg} = 3820 \text{ kg} \]
03

Apply conservation of momentum principle

The principle of conservation of translational momentum states that the total momentum of a closed system remains constant if no external forces act on it. The initial momentum is given by \[ p_i = m_i v_i \] where \( m_i \) is the initial mass and \( v_i \) is the initial velocity. The final momentum is given by \[ p_f = m_f v_f \] where \( m_f \) is the final mass and \( v_f \) is the final velocity. Since momentum is conserved, \[ p_i = p_f \]
04

Set up the momentum equation

Substitute the values into the momentum equation: \[ m_i v_i = m_f v_f \] \[ 2900 \text{ kg} \times 250 \text{ m/s} = 3820 \text{ kg} \times v_f \]
05

Solve for the final velocity

Rearrange the equation to solve for the final velocity \( v_f \): \[ v_f = \frac{2900 \text{ kg} \times 250 \text{ m/s}}{3820 \text{ kg}} \] Perform the calculation: \[ v_f \approx 189.53 \text{ m/s} \]

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Key Concepts

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

conservation of momentum
Conservation of momentum is a fundamental concept in physics. It states that the total momentum of a closed system remains constant if no external forces act on it.
Momentum is defined as the product of mass and velocity, represented by the formula: \(p = mv\).

In the rocket sled problem, the system consists of the sled and the scooped water. Initially, the sled has a certain momentum, calculated using its mass and velocity. When the sled scoops up water, the total mass changes, but the system's momentum must remain constant because no external force is acting on it.
This principle allows us to determine the new speed (final velocity) of the sled after the water has been scooped up.
translational motion
Translational motion refers to the movement of an object in space, where every part of the object moves together in the same direction. It is different from rotational motion, where parts of the object move around an axis.
In the context of the rocket sled problem, we only consider translational motion. The sled moves along the tracks in a straight line, and its velocity and mass determine how it changes when scooping up water.

This type of motion is governed by Newton's laws of motion, particularly the conservation of momentum, which we use to solve for the sled's speed after the water is added.
kinematics
Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. It focuses on parameters such as displacement, velocity, and acceleration.

In the rocket sled problem, we primarily deal with initial and final velocities, as well as changes in mass, to determine the sled's speed after scooping water.
The motion along the track can be described through these kinematic parameters, and by using conservation laws, we find the final velocity of the sled.
physics problems
Physics problems often involve applying fundamental principles to real-world scenarios. The rocket sled question is a perfect example.
By breaking down the problem into smaller, manageable steps, we use basic concepts such as conservation of momentum and kinematics to arrive at a solution.
This type of problem-solving helps bridge theoretical physics and practical applications, enhancing understanding and skill in applying physics principles to various situations.
momentum calculation
Calculating momentum involves multiplying the mass of an object by its velocity. This gives a measure of the object's motion. For the rocket sled problem:
  • Initial momentum: \(p_i = m_i v_i \)
  • Final momentum: \(p_f = m_f v_f \)
    Given that momentum is conserved, we set these two equal, allowing us to solve for the unknown final velocity.

    Using the given values, the calculation simplifies to:
    \[ v_f = \frac{2900 \text{ kg} \times 250 \text{ m/s}}{3820 \text{ kg}} \]
    Resulting in a final speed: \(v_f \approx 189.53 \text{ m/s} \)

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Most popular questions from this chapter

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