Problem 94 - Thomas Calculus

Question

Liftoff from Earth 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 \(\mathrm{min}\) later?

Step-by-Step Solution

Answer

The rocket will be traveling at 1200 m/s after 1 minute.

Step 1: Understand the Given Values

Identify the values given in the problem. The acceleration of the rocket is 20 meters per second squared, and the time duration is 1 minute, which needs to be converted into seconds: 1 minute = 60 seconds.

Step 2: Use the Formula to Calculate Final Velocity

The formula to find the final velocity when an object starts from rest and accelerates with a constant rate is given by: \[ v = u + a \cdot t \]where:- \( v \) is the final velocity,- \( u \) is the initial velocity (0 m/s because the rocket starts from rest),- \( a \) is the acceleration (20 m/s²),- \( t \) is time (60 seconds).

Step 3: Plug in the Values and Solve

Substitute the known values into the formula:\[ v = 0 + 20 \times 60 \]Calculate the product:\[ v = 0 + 1200 = 1200 \text{ m/s} \]

Step 4: Conclusion: Final Velocity of the Rocket

The final velocity of the rocket after 1 minute of constant acceleration at 20 m/s² is 1200 m/s.

Key Concepts

Understanding Acceleration

Acceleration is a fundamental concept in physics that describes how the velocity of an object changes over time. It is important to remember that acceleration can occur in the form of speeding up, slowing down, or changing direction.
In the context of the rocket liftoff problem, acceleration is specified as 20 meters per second squared (m/s²). This indicates that every second, the rocket's velocity increases by 20 meters per second.

• Acceleration is a vector quantity, which means it has both magnitude and direction.
• Constant acceleration means the rate of change of velocity remains the same during the time period.
• In calculations, acceleration is usually denoted as \( a \).

Understanding acceleration helps in predicting how the velocity of an object will change, which is crucial for tasks such as launching rockets.

Mastering Velocity Calculation

Velocity calculation is the process of determining the final velocity of an object when its initial velocity, acceleration, and time are known. This is particularly important for objects in motion under the influence of constant forces.
A pivotal formula in physics for calculating velocity is:
\[ v = u + a \cdot t \]
where:

• \( v \) is the final velocity.
• \( u \) is the initial velocity, which can be zero if the object starts from rest.
• \( a \) is the acceleration, measured in \( \mathrm{m/s^2} \).
• \( t \) is the time during which acceleration is applied.

The exercise uses this formula to find the rocket's velocity after a given time with constant acceleration. By substituting the known values of initial velocity, acceleration, and time into this formula, the final velocity can be computed easily.

Exploring Rocket Motion

Rocket motion involves understanding how rockets move and are propelled, often involving concepts like Newton's laws of motion and the physics of acceleration.
In the realm of rocket science, a significant focus is on achieving liftoff while counteracting Earth's gravity, which requires a powerful thrust and acceleration.
For the exercise:

• The rocket starts from rest, implying an initial velocity of zero.
• Constant acceleration is employed to gradually increase velocity, allowing the rocket to overcome gravitational pull.
• By knowing the rocket's acceleration, we can predict its velocity after a certain period, which is critical for ensuring successful launch trajectories.

Rocket motion exemplifies the fundamental principles of acceleration and velocity on a grand scale, where precise calculations ensure successful travel beyond Earth's atmosphere.