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Chapter 4: Problem 124

Liftoff from Earth \(\quad\) A rocket lifts off the surface of Earth with a constant acceleration of \(20 \mathrm{m} / \mathrm{sec}^{2} .\) How fast will the rocket be going 1 min later?

Short Answer

Expert verified
The rocket will be going 1200 m/s after 1 minute.

Step by step solution

01

Identify Given Values and Required Output

The rocket's acceleration is given as \( a = 20 \text{ m/s}^2 \). The time duration for which the rocket accelerates is \( t = 1 \text{ min} = 60 \text{ sec} \). We are required to find the final velocity \( v_f \) of the rocket after 60 seconds.
02

Understand the Relevant Equation

The equation that relates initial velocity, acceleration, and time to the final velocity is \( v_f = v_i + a \cdot t \), where \( v_i \) is the initial velocity, \( a \) is the acceleration, and \( t \) is time.
03

Initial Conditions

Since the rocket lifts off from rest, the initial velocity \( v_i \) is \( 0 \text{ m/s} \).
04

Plug in the Values to Calculate Final Velocity

Substitute \( v_i = 0 \text{ m/s} \), \( a = 20 \text{ m/s}^2 \), and \( t = 60 \text{ sec} \) into the equation \( v_f = v_i + a \cdot t \). Hence, \( v_f = 0 + 20 \times 60 \).
05

Compute the Final Velocity

Calculate \( v_f = 20 \times 60 = 1200 \text{ m/s} \). This is the final velocity of the rocket after 1 minute.

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Key Concepts

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

Constant Acceleration
Acceleration is a key concept in kinematics, especially when studying the motion of rockets. When an object accelerates at a constant rate, it means its speed increases by the same amount every second.
For a rocket lifting off, this implies that every second, the rocket's speed will increase by the amount of its constant acceleration. How does this work? Let's ponder a little. Constant acceleration means no sways or drops in its value over time.
  • In our example, the constant acceleration is given as 20 m/s虏.
  • This means that every second, the speed increases by 20 meters per second.
  • Consistency in acceleration is crucial in achieving a steady increase in velocity when a rocket launches.
The concept of constant acceleration allows predictions on how fast the rocket will travel over a given time period.
Final Velocity Calculation
To calculate the final velocity when dealing with constant acceleration, we use a specific formula. The goal is to discover how fast the object, like our rocket, is moving after a given time. The formula to find final velocity (\( v_f \)) is:\[v_f = v_i + a \cdot t\]Where:
  • \( v_f \) is the final velocity.
  • \( v_i \) is the initial velocity.
  • \( a \) is the acceleration (in m/s虏).
  • \( t \) is time (in seconds).
In our scenario:
  • Initial velocity \( v_i \) is 0 m/s (as the rocket starts from rest).
  • Acceleration \( a \) is 20 m/s虏.
  • Time \( t \) is converted from 1 minute to 60 seconds.
The calculation then becomes a matter of plugging these numbers into the equation: \[v_f = 0 + 20 \times 60 = 1200 \text{ m/s}\]The rocket reaches 1200 meters per second after 60 seconds of accelerating.
Rockets
Rockets are fascinating due to their powerful propulsion systems and ability to conquer gravity. They rely on a fundamental principle of physics: Newton's Third Law of Motion, which states that for every action, there is an equal and opposite reaction.
This principle is exemplified in the way a rocket lifts off, overcoming Earth's gravitational pull through propulsion. Here are key aspects to consider about rockets:
  • Propulsion: Combustion of fuel creates high-pressure gases expelled downward, pushing the rocket upward.
  • Acceleration: Once a rocket starts, it often accelerates at a constant rate to reach optimal velocities needed for its journey.
  • Launch Phases: Rockets often operate in stages, discarding parts to become lighter and maintain successful propulsion as they ascend.
Understanding rockets involves grasping these concepts of force, motion, and engineering marvel, making them truly impressive feats of technology.

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