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Chapter 4: Problem 84

A small rocket-propelled vehicle weighs \(125 \mathrm{lb}\), in cluding 20 lb of fuel. Fuel is burned at the constant rate of 2 lb/sec with an exhaust velocity relative to the nozzle of \(400 \mathrm{ft} / \mathrm{sec}\). Upon ignition the vehicle is released from rest on the \(10^{\circ}\) incline. Calculate the maximum velocity \(v\) reached by the vehicle. Neglect all friction.

Short Answer

Expert verified
The maximum velocity reached by the vehicle is approximately 69.76 ft/sec.

Step by step solution

01

Determine Initial and Final Mass of the Vehicle

The total initial mass of the vehicle, including fuel, is 125 lb. The fuel contributes 20 lb to this mass. At maximum velocity, all fuel will be burned, so the final mass of the vehicle is 105 lb (125 lb - 20 lb).
02

Define Relevant Forces and Acceleration

As the vehicle is inclined at a 10掳 angle, the component of gravitational force along the slope is given by \( F_g = mg \sin \theta \). Converting the initial mass from pounds to mass (slug) using \( g = 32.2 \ \text{ft/s}^2 \), we have \( m = \frac{125}{32.2} \ \text{slug} \).
03

Apply the Rocket Equation

The rocket equation is given by \( v = v_{e} \ln \left( \frac{m_i}{m_f} \right) \), where \(v_e\) is the effective exhaust velocity (400 ft/sec), and \(m_i\) and \(m_f\) are the initial mass and the final mass of the vehicle, respectively. Insert these values into the equation to get \( v = 400 \ln \left( \frac{125}{105} \right) \).
04

Calculate Maximum Velocity

Compute the maximum velocity by substituting values into the rocket equation: \( v = 400 \ln \left( \frac{125}{105} \right) = 400 \ln(1.1905) \). Calculating this gives \( v \approx 400 \times 0.1744 \approx 69.76 \text{ ft/sec} \).
05

Consider Incline Effect (Neglect as the vehicle is still under rocket propulsion)

Even though the vehicle is on an incline, we focus only on the rocket propulsion for this calculation since we assumed no friction and no initial velocity to overcome before the rocket operates.

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Key Concepts

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

Rocket Propulsion
Rocket propulsion is the force that moves rockets and propelled vehicles forward by ejecting gas at high speed in the opposite direction. This principle is based on Newton's Third Law of Motion, which states that for every action, there is an equal and opposite reaction.
For a rocket-propelled vehicle, the fuel burns inside the engine to produce hot gases. These gases escape through the rocket nozzle, providing thrust and causing the vehicle to accelerate.
Rocket propulsion is influenced by factors such as:
  • The rate of fuel consumption
  • The exhaust velocity of the escaping gases
  • The angle of movement (like inclines)
These factors combined determine how efficiently and quickly a rocket can reach its intended speed and direction.
Rocket Equation
The rocket equation, also known as the Tsiolkovsky rocket equation, is fundamental in calculating a rocket's motion. It connects several key variables to predict how the rocket's velocity changes as fuel is expended.
Mathematically, the rocket equation is represented as:
  • \[ v = v_e \, \ln \left( \frac{m_i}{m_f} \right) \]
In this equation:
  • \( v \) is the change in velocity of the rocket.
  • \( v_e \) is the effective exhaust velocity, a measure of the speed at which exhaust leaves the rocket.
  • \( m_i \) and \( m_f \) represent the initial and final mass of the vehicle, respectively.
This formula indicates that the faster the exhaust velocity and the greater the mass of burnt fuel, the higher the velocity the rocket can achieve.
Exhaust Velocity
Exhaust velocity is a crucial component in understanding rocket propulsion and is defined as the speed at which combustion gases exit the rocket engine.
It plays a significant role in determining a rocket's thrust and its efficiency, as seen in the rocket equation.
The higher the exhaust velocity, the greater the rocket's thrust for a given rate of fuel consumption. This means rockets with more efficient engines (higher exhaust velocity) can achieve higher velocities, or the same velocity with less fuel. In our problem, the exhaust velocity is given as 400 ft/s, which is used in calculations to determine the vehicle's final velocity when all fuel is consumed.
Inclined Plane
An inclined plane is a flat surface tilted at an angle, like a slope, which can affect the motion of objects placed on it.
In the context of our rocket-propelled vehicle, the incline affects the force and acceleration experienced by the vehicle.
However, in this particular case, the effects of the incline are neglected for simplicity.
Typically, on an incline:
  • Gravity causes a force component parallel to the plane, affecting acceleration
  • The angle of incline can reduce or increase the effective gravitational pull opposing the vehicle's motion
Understanding inclined planes is important in physics, especially while analyzing forces and motion for vehicles on non-flat surfaces.

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Most popular questions from this chapter

The man of mass \(m_{1}\) and the woman of mass \(m_{2}\) are standing on opposite ends of the platform of mass \(m_{0}\) which moves with negligible friction and is initially at rest with \(s=0 .\) The man and woman begin to approach each other. Derive an expression for the displacement \(s\) of the platform when the two meet in terms of the displacement \(x_{1}\) of the man relative to the platform.

At a bulk loading station, gravel leaves the hopper at the rate of 220 lb/sec with a velocity of \(10 \mathrm{ft} / \mathrm{sec}\) in the direction shown and is deposited on the moving flatbed truck. The tractive force between the driving wheels and the road is \(380 \mathrm{lb}\), which overcomes the 200 lb of frictional road resistance. Determine the acceleration \(a\) of the truck 4 seconds after the hopper is opened over the truck bed, at which instant the truck has a forward speed of \(1.5 \mathrm{mi} / \mathrm{hr}\) The empty weight of the truck is \(12,000 \mathrm{lb}\).

A centrifuge consists of four cylindrical containers, each of mass \(m\), at a radial distance \(r\) from the rotation axis. Determine the time \(t\) required to bring the centrifuge to an angular velocity \(\omega\) from rest under a constant torque \(M\) applied to the shaft. The diameter of each container is small compared with \(r,\) and the mass of the shaft and supporting arms is small compared with \(m\).

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The small car, which has a mass of \(20 \mathrm{kg},\) rolls freely on the horizontal track and carries the \(5-\mathrm{kg}\) sphere mounted on the light rotating rod with \(r=\) \(0.4 \mathrm{m} .\) A geared motor drive maintains a constant angular speed \(\dot{\theta}=4\) rad/s of the rod. If the car has a velocity \(v=0.6 \mathrm{m} / \mathrm{s}\) when \(\theta=0,\) calculate \(v\) when \(\theta=60^{\circ}\). Neglect the mass of the wheels and any friction.
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