Problem 98 - Thomas Calculus in SI Units

Question

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

Step-by-Step Solution

Answer

The rocket's velocity after 1 minute is 1200 m/s.

Step 1: Identify the Known Values

First, we need to identify the known values given in the problem. The initial velocity of the rocket is 0 m/s because it starts from rest. The acceleration of the rocket is given as 20 m/s². The time during which the rocket accelerates is 1 minute, which is equivalent to 60 seconds.

Step 2: Use the Equation of Motion

The equation of motion that relates initial velocity, acceleration, and time to final velocity is \( v = u + at \), where \( v \) is the final velocity, \( u \) is the initial velocity, \( a \) is the acceleration, and \( t \) is the time in seconds.

Step 3: Plug in the Values

Substitute the known values into the equation: \( u = 0 \), \( a = 20 \ \mathrm{m/s²} \), and \( t = 60 \ \mathrm{s} \). This gives us:\[ v = 0 + (20 \ \mathrm{m/s²})(60 \ \mathrm{s}) \].

Step 4: Calculate the Final Velocity

Perform the multiplication to find the final velocity: \( v = (20 \ \mathrm{m/s²})(60 \ \mathrm{s}) = 1200 \ \mathrm{m/s} \). Thus, the final velocity of the rocket after 1 minute is 1200 m/s.

Key Concepts

Final Velocity

The concept of final velocity is pivotal in understanding how objects move through space and time. In this particular context, final velocity refers to the speed that a rocket achieves after accelerating from a standstill during a given time period. To determine the final velocity, it is essential to consider the initial speed of the object, which in this exercise, is 0 m/s because the rocket starts from rest.

Final velocity is influenced by two key factors:

• The constant acceleration that the object is subjected to.
• The duration of time over which the object accelerates.

By manipulating these variables, we can predict the final speed of an object using the equation of motion.

Constant Acceleration

Constant acceleration occurs when an object's velocity changes at a steady rate over time. This means that the speed of the object increases equally in every second it is accelerating. For the rocket lifting off from Earth, the given constant acceleration is 20 m/s².

Understanding constant acceleration is crucial because it simplifies the calculations involved in motion problems. With constant acceleration, the motion of an object can be easily predicted using standard equations of motion.

• This type of acceleration implies that there are no variations in the rate of increase of velocity during the time frame considered.
• The object experiences a uniform increase in speed each second it moves.

With this knowledge, solving problems to find parameters like velocity or distance becomes straightforward.

Equation of Motion

The equation of motion used in this problem is a standard tool in kinematics for predicting the final velocity of a moving object. The formula used is:
\[ v = u + at \]where:

• \( v \) represents the final velocity we want to find.
• \( u \) stands for the initial velocity.
• \( a \) indicates the constant acceleration.
• \( t \) denotes the time of acceleration.

For instance, in the given example, the initial velocity \( u \) is 0 m/s, the constant acceleration \( a \) is 20 m/s², and the time \( t \) is 60 seconds (1 minute). By substituting these values into the formula, we can calculate the rocket's final velocity after 1 minute, which is 1200 m/s. The equation of motion is a vital concept because it offers a simple means of predicting movement in physics.