/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 18 A \(523-\mathrm{kg}\) experiment... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 18

A \(523-\mathrm{kg}\) experimental rocket sled can be accelerated from rest to \(1620 \mathrm{~km} / \mathrm{h}\) in \(1.82 \mathrm{~s}\). What net force is required?

Short Answer

Expert verified
The net force required to accelerate the rocket sled is approximately \(129308 \, \text{N}\).

Step by step solution

01

Identifying and Converting Units

Identify the given variables: mass (m) = 523 kg, initial velocity (u) = 0 km/h, final velocity (v) = 1620 km/h, and time (t) = 1.82 s. For consistent units in the upcoming calculations, convert the final velocity from km/h to m/s: \(1620 \, \text{km/h} = 1620 \times \, \frac{1000}{3600} \, \text{m/s} = 450 \, \text{m/s}\).
02

Calculating Acceleration

Accelerations (a) is the change in velocity per unit time. The formula is \(a = \frac{v - u}{t}\). Substituting the values, we get: \(a = \frac{450 \, \text{m/s} - 0}{1.82 \, \text{s}} = 247.25 \, \text{m/s}^2\).
03

Calculating Force

Force (F) can be calculated using Newton's second law of motion: \(F = m \times a\). This gives: \(F = 523 \, \text{kg} \times 247.25 \, \text{m/s}^2 = 129307.75 \, \text{N}\).

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Key Concepts

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

Force Calculation
Newton's Second Law of Motion is key to understanding force calculation. It is expressed as:
\[ F = m \times a \]where:
  • \( F \) is the net force applied, measured in Newtons (N)
  • \( m \) is the mass of the object in kilograms (kg)
  • \( a \) is the acceleration in meters per second squared (m/s虏)
To determine the force required to accelerate an object, multiply its mass by the acceleration it experiences. Remember, the force is the cause of the acceleration sensed by the object. So, in our example, a 523 kg rocket sled accelerating at 247.25 m/s虏 requires a force of 129,307.75 N. This means a mighty push is necessary to overcome inertia and speed up the sled that quickly.
Unit Conversion
In physics, consistency in units simplifies calculations and prevents errors. Often, problems provide measurements in various units, requiring conversion to be compatible with equations. One common conversion is between kilometers per hour (km/h) and meters per second (m/s). Using the relation:
\[ 1 \, \text{km/h} = \frac{1000}{3600} \, \text{m/s} = \frac{5}{18} \, \text{m/s} \]Apply this to convert final velocities from km/h to m/s. So, a velocity of 1620 km/h becomes 450 m/s after conversion. These conversions ensure that all variables are expressed in compatible units, crucial for correct application of physics formulas, like those in Newton's laws.
Acceleration
Acceleration is the rate at which an object's velocity changes over time. It is defined by the equation:
\[ a = \frac{v - u}{t} \]where:
  • \( v \) is the final velocity
  • \( u \) is the initial velocity
  • \( t \) is the time period over which the change occurs
In our scenario, the initial velocity is 0 m/s, and the final velocity is 450 m/s, achieved over 1.82 seconds. Substituting these values, the acceleration is calculated as 247.25 m/s虏. High acceleration indicates how rapidly the rocket sled increases speed. Understanding acceleration helps grasp how quickly a body can change its state of motion.

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Most popular questions from this chapter

(a) Neglecting gravitational forces, what force would be required to accelerate a 1200-metric-ton spaceship from rest to one-tenth the speed of light in 3 days? In 2 months? (One metric ton \(=1000 \mathrm{~kg} .)(b)\) Assuming that the engines are shut down when this speed is reached, what would be the time required to complete a 5-light-month journey for each of these two cases? (Use 1 month \(=30\) days.)

A space traveler whose mass is \(75.0 \mathrm{~kg}\) leaves Earth. Compute his weight \((a)\) on Earth, \((b)\) on Mars, where \(g=3.72\) \(\mathrm{m} / \mathrm{s}^{2}\), and \((c)\) in interplanetary space. \((d)\) What is his mass at each of these locations?

A rocket and its payload have a total mass of \(51,000 \mathrm{~kg} .\) How large is the thrust of the rocket engine when \((a)\) the rocket is "hovering" over the launch pad, just after ignition, and (b) when the rocket is accelerating upward at \(18 \mathrm{~m} / \mathrm{s}^{2} ?\)

A man of mass \(83 \mathrm{~kg}\) (weight \(180 \mathrm{lb}\) ) jumps down to a concrete patio from a window ledge only \(0.48 \mathrm{~m}(=1.6 \mathrm{ft})\) above the ground. He neglects to bend his knees on landing, so that his motion is arrested in a distance of about \(2.2 \mathrm{~cm}(=0.87\) in.). ( \(a\) ) What is the average acceleration of the man from the time his feet first touch the patio to the time he is brought fully to rest? ( \(b\) ) With what average force does this jump jar his bone structure?

A 26-ton Navy jet (Fig. 3-27) requires an air speed of \(280 \mathrm{ft} / \mathrm{s}\) for lift-off. Its own engine develops a thrust of \(24,000 \mathrm{lb}\). The jet is to take off from an aircraft carrier with a \(300-\mathrm{ft}\) flight deck. What force must be exerted by the catapult of the carrier? Assume that the catapult and the jet's engine each exert a constant force over the 300 -ft takeoff distance.
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