Problem 43 - College Physics

Question

Two rockets having the same acceleration start from rest, but rocket \(A\) travels for twice as much time as rocket \(B\). (a) If rocket \(A\) goes a distance of \(250 \mathrm{~km}\), how far will rocket \(B\) go? (b) If rocket \(A\) reaches a speed of \(350 \mathrm{~m} / \mathrm{s},\) what speed will rocket \(B\) reach?

Step-by-Step Solution

Answer

Rocket B travels 125 km and reaches a speed of 175 m/s.

Step 1: Understanding the Problem

We are given two rockets, A and B, with the same acceleration. Rocket A travels for twice the time rocket B does. We need to find the distance rocket B travels if A travels 250 km, and the speed B reaches if A reaches 350 m/s.

Step 2: Using the Distance Formula

To find the distance traveled by rocket B, use the kinematic equation: \[ d = \frac{1}{2} a t^2 \]For rocket A, \(d_A = \frac{1}{2} a (2t)^2 = 2at^2\).Given \(d_A = 250 \text{ km}\), we can find that: \[ 2at^2 = 250 \times 10^3 \text{ m} \].

Step 3: Solving for Rocket B's Distance

Rocket B's distance \(d_B = \frac{1}{2} a t^2\).Using the result from Step 2: \[ d_B = \frac{1}{2} (2at^2) / 2 = \frac{250 \times 10^3}{2} = 125 \text{ km} \].So, rocket B travels 125 km.

Step 4: Using the Final Velocity Formula

To find the speed of rocket B, use the kinematics equation: \[ v = at \].For rocket A, \(v_A = a(2t) = 2at\).Given \(v_A = 350 \text{ m/s}\), we express it as \[ 2at = 350 \].

Step 5: Solving for Rocket B's Speed

Rocket B's speed \(v_B = at\).From the result in Step 4: \[ v_B = \frac{350}{2} = 175 \text{ m/s} \].Thus, rocket B reaches a speed of 175 m/s.

Key Concepts

Acceleration

Acceleration is a fundamental concept in kinematics, which is the branch of physics that describes the motion of objects. It refers to the rate at which an object changes its velocity.
Acceleration is not just about speeding up; it can also involve slowing down or changing direction. It is a vector quantity, which means it has both magnitude and direction.

• A constant acceleration implies that the object's velocity is changing at a steady rate over time.
• If two objects have the same acceleration, as in the case of rockets A and B, they have the same change in velocity over equal time intervals.
• The formula for acceleration is: \[ a = \frac{\Delta v}{\Delta t} \] where \( \Delta v \) is the change in velocity and \( \Delta t \) is the change in time.

Understanding acceleration is crucial to solving problems involving motion, such as determining how far or how fast an object will travel over a period of time.

Distance Formula

The distance formula in kinematics helps us calculate how far an object travels under constant acceleration. For an object starting from rest with uniform acceleration, we can use one of the kinematic equations to calculate the distance:
\[d = \frac{1}{2} a t^2\]
This formula originates from the integration of the acceleration function, accounting for the initial velocity of zero. The simplicity of the equation reflects how distance depends quadratically on time while under constant acceleration.

• When the time doubles, the distance increases by four times, assuming acceleration stays constant. This is because distance is proportional to the square of time.
• In our problem, rocket A travels for twice as long as rocket B, which significantly impacts the distance covered because time appears squared in the formula.

Understanding this relationship helps in predicting how changes in time affect the distance traveled under constant acceleration.

Final Velocity

Final velocity is a key concept in understanding the motion of an object under constant acceleration. It signifies the speed an object reaches after accelerating for a specific period of time.
Using the kinematic equation for velocity: \[ v = at \]
we can derive the final velocity of an object starting from rest. Here, \( v \) is the final velocity, \( a \) is the acceleration, and \( t \) is the time period for which it accelerates.

• For example, rocket A's final velocity is achieved after accelerating for twice the time of rocket B. Thus, its velocity is twice that of rocket B, assuming the accelerations are the same.
• Understanding how final velocity is calculated is crucial for analyzing motion parameters like speed and distance over time.

The final velocity not only tells us about the current speed of an object but also gives insights into the dynamics of its motion.

Kinematic Equations

Kinematic equations are a set of equations that describe the motion of objects under constant acceleration. These equations are essential tools for solving problems involving linear motion.
The basic kinematic equations include:

• \[ v = u + at \]li>\[ d = ut + \frac{1}{2} at^2 \]
• \[ v^2 = u^2 + 2ad \]

where \( u \) is the initial velocity, \( v \) is the final velocity, \( a \) is the acceleration, \( d \) is the distance, and \( t \) is the time.

• These equations relate all the motion variables and help in predicting one variable when others are known.
• In the problem with the rockets, these equations allow us to find the distance and velocity by using time, given the rockets start from rest (\( u = 0 \)).

Understanding and using kinematic equations enable us to solve a wide range of motion problems and are foundational in the study of physics.