Determining the Time Required for the Boat to Travel a Distance of 400 m

Oct 13, 2025 by wbrum.org 107 views

Understanding the Problem

When a boat travels along a straight line, its motion can be described by a graph that shows its speed over time. In this scenario, we are given a graph that represents the speed of the boat as a function of time. Our task is to construct two important graphs: the s-t graph, which shows the distance traveled by the boat over time, and the a-s graph, which represents the acceleration of the boat as a function of its speed.

Constructing the s-t Graph

To construct the s-t graph, we need to understand the relationship between distance, speed, and time. The distance traveled by the boat is given by the equation:

s = ∫v(t)dt

where s is the distance, v(t) is the speed at time t, and the integral is taken over the time interval from 0 to t.

Since we are given the speed graph, we can use it to find the distance traveled by the boat at any given time. We can do this by integrating the speed graph over time.

Let's assume that the speed graph is a straight line with a constant speed of 5 m/s for the first 10 seconds, and then it increases linearly to 10 m/s over the next 10 seconds. After that, it remains constant at 10 m/s.

To find the distance traveled by the boat, we can integrate the speed graph over time. For the first 10 seconds, the distance traveled is:

s1 = ∫(5 m/s)dt = 5t | 0 to 10 = 50 m

For the next 10 seconds, the speed increases linearly from 5 m/s to 10 m/s. We can find the distance traveled during this time by integrating the average speed, which is (5 + 10)/2 = 7.5 m/s.

s2 = ∫(7.5 m/s)dt = 7.5t | 10 to 20 = 75 m

After 20 seconds, the speed remains constant at 10 m/s. We can find the distance traveled during this time by integrating the speed graph.

s3 = ∫(10 m/s)dt = 10t | 20 to 30 = 100 m

The total distance traveled by the boat is the sum of the distances traveled during each time interval:

s = s1 + s2 + s3 = 50 + 75 + 100 = 225 m

Constructing the a-s Graph

To construct the a-s graph, we need to understand the relationship between acceleration, speed, and time. The acceleration of the boat is given by the equation:

a = dv/dt

where a is the acceleration, v is the speed, and the derivative is taken with respect to time.

Since we are given the speed graph, we can use it to find the acceleration of the boat at any given time. We can do this by taking the derivative of the speed graph with respect to time.

To find the acceleration of the boat, we can take the derivative of the speed graph with respect to time. For the first 10 seconds, the acceleration is:

a1 = dv/dt = 0

For the next 10 seconds, the speed increases linearly from 5 m/s to 10 m/s. We can find the acceleration during this time by taking the derivative of the speed graph.

a2 = dv/dt = (10 - 5)/10 = 0.5 m/s^2

After 20 seconds, the speed remains constant at 10 m/s. We can find the acceleration during this time by taking the derivative of the speed graph.

a3 = dv/dt = 0

The acceleration of the boat is the sum of the accelerations during each time interval:

a = a1 + a2 + a3 = 0 + 0.5 + 0 = 0.5 m/s^2

Determining the Time Required for the Boat to Travel a Distance of 400 m

To determine the time required for the boat to travel a distance of 400 m, we need to use the equation:

s = ∫v(t)dt

where s is the distance, v(t) is the speed at time t, and the integral is taken over the time interval from 0 to t.

We can use the speed graph to find the time required for the boat to travel a distance of 400 m. We can do this by integrating the speed graph over time until the distance traveled is equal to 400 m.

To find the time required for the boat to travel a distance of 400 m, we can integrate the speed graph over time. For the first 10 seconds, the distance traveled is:

s1 = ∫(5 m/s)dt = 5t | 0 to 10 = 50 m

For the next 10 seconds, the speed increases linearly from 5 m/s to 10 m/s. We can find the distance traveled during this time by integrating the average speed, which is (5 + 10)/2 = 7.5 m/s.

s2 = ∫(7.5 m/s)dt = 7.5t | 10 to 20 = 75 m

After 20 seconds, the speed remains constant at 10 m/s. We can find the distance traveled during this time by integrating the speed graph.

s3 = ∫(10 m/s)dt = 10t | 20 to 30 = 100 m

The total distance traveled by the boat is the sum of the distances traveled during each time interval:

s = s1 + s2 + s3 = 50 + 75 + 100 = 225 m

Since the boat has traveled 225 m in 30 seconds, we can find the time required for the boat to travel an additional 175 m (400 - 225) by integrating the speed graph over time.

s4 = ∫(10 m/s)dt = 10t | 30 to 40 = 100 m

The boat has traveled an additional 100 m in 10 seconds. We can find the time required for the boat to travel the remaining 75 m (175 - 100) by integrating the speed graph over time.

s5 = ∫(10 m/s)dt = 10t | 40 to 50 = 75 m

The boat has traveled the remaining 75 m in 10 seconds. Therefore, the total time required for the boat to travel a distance of 400 m is:

t = 30 + 10 + 10 = 50 seconds

Conclusion

In this article, we constructed the s-t and a-s graphs for a boat traveling along a straight line with a speed described by a graph. We also determined the time required for the boat to travel a distance of 400 m. The s-t graph shows the distance traveled by the boat over time, while the a-s graph represents the acceleration of the boat as a function of its speed. The time required for the boat to travel a distance of 400 m is 50 seconds.

Q: What is the relationship between distance, speed, and time?

A: The distance traveled by an object is given by the equation:

s = ∫v(t)dt

where s is the distance, v(t) is the speed at time t, and the integral is taken over the time interval from 0 to t.

Q: How do you construct the s-t graph?

A: To construct the s-t graph, you need to integrate the speed graph over time. This will give you the distance traveled by the object at any given time.

Q: What is the relationship between acceleration, speed, and time?

A: The acceleration of an object is given by the equation:

a = dv/dt

where a is the acceleration, v is the speed, and the derivative is taken with respect to time.

Q: How do you construct the a-s graph?

A: To construct the a-s graph, you need to take the derivative of the speed graph with respect to time. This will give you the acceleration of the object at any given time.

Q: What is the time required for the boat to travel a distance of 400 m?

A: To determine the time required for the boat to travel a distance of 400 m, you need to integrate the speed graph over time until the distance traveled is equal to 400 m. In this case, the time required is 50 seconds.

Q: What is the acceleration of the boat?

A: The acceleration of the boat is given by the equation:

a = dv/dt

where a is the acceleration, v is the speed, and the derivative is taken with respect to time. In this case, the acceleration of the boat is 0.5 m/s^2.

Q: What is the speed of the boat at any given time?

A: The speed of the boat is given by the graph, which shows a constant speed of 5 m/s for the first 10 seconds, and then it increases linearly to 10 m/s over the next 10 seconds. After that, it remains constant at 10 m/s.

Q: What is the distance traveled by the boat at any given time?

A: The distance traveled by the boat is given by the equation:

s = ∫v(t)dt

where s is the distance, v(t) is the speed at time t, and the integral is taken over the time interval from 0 to t.

Q: How do you determine the time required for the boat to travel a certain distance?

A: To determine the time required for the boat to travel a certain distance, you need to integrate the speed graph over time until the distance traveled is equal to the desired distance.

Q: What is the relationship between the s-t graph and the a-s graph?

A: The s-t graph shows the distance traveled by the boat over time, while the a-s graph represents the acceleration of the boat as a function of its speed.

Q: How do you use the s-t graph and the a-s graph to determine the time required for the boat to travel a certain distance?

A: You can use the s-t graph to determine the distance traveled by the boat at any given time, and the a-s graph to determine the acceleration of the boat at any given time. By combining this information, you can determine the time required for the boat to travel a certain distance.

Q: What are some common applications of the s-t graph and the a-s graph?

A: The s-t graph and the a-s graph are commonly used in physics and engineering to describe the motion of objects. They are used to determine the distance traveled by an object, the acceleration of an object, and the time required for an object to travel a certain distance.

Q: How do you interpret the s-t graph and the a-s graph?

A: The s-t graph shows the distance traveled by the boat over time, while the a-s graph represents the acceleration of the boat as a function of its speed. By interpreting these graphs, you can gain a better understanding of the motion of the boat and make predictions about its future behavior.

Q: What are some common mistakes to avoid when working with the s-t graph and the a-s graph?

A: Some common mistakes to avoid when working with the s-t graph and the a-s graph include:

Failing to take into account the initial conditions of the problem
Failing to consider the limitations of the graph
Failing to use the correct units
Failing to check the accuracy of the graph

Q: How do you use the s-t graph and the a-s graph to solve problems?

A: To use the s-t graph and the a-s graph to solve problems, you need to:

Identify the problem and the relevant information
Determine the type of motion (e.g. constant speed, acceleration, etc.)
Use the s-t graph to determine the distance traveled by the object
Use the a-s graph to determine the acceleration of the object
Combine this information to determine the time required for the object to travel a certain distance.

Q: What are some real-world applications of the s-t graph and the a-s graph?

A: The s-t graph and the a-s graph have many real-world applications, including:

Describing the motion of objects in physics and engineering
Predicting the behavior of objects in different situations
Designing and optimizing systems
Analyzing data and making predictions about future behavior.

Q: How do you create a s-t graph and an a-s graph?

A: To create a s-t graph and an a-s graph, you need to:

Determine the type of motion (e.g. constant speed, acceleration, etc.)
Use a graphing tool or software to create the graph
Plot the distance traveled by the object over time (s-t graph)
Plot the acceleration of the object as a function of its speed (a-s graph)

Q: What are some common tools and software used to create s-t graphs and a-s graphs?

A: Some common tools and software used to create s-t graphs and a-s graphs include:

Graphing calculators
Computer software (e.g. MATLAB, Python, etc.)
Online graphing tools (e.g. Desmos, etc.)

Q: How do you interpret the results of a s-t graph and an a-s graph?

A: To interpret the results of a s-t graph and an a-s graph, you need to:

Understand the type of motion (e.g. constant speed, acceleration, etc.)
Identify the key features of the graph (e.g. distance traveled, acceleration, etc.)
Use the graph to make predictions about the future behavior of the object.

Q: What are some common challenges when working with s-t graphs and a-s graphs?

A: Some common challenges when working with s-t graphs and a-s graphs include:

Failing to take into account the initial conditions of the problem
Failing to consider the limitations of the graph
Failing to use the correct units
Failing to check the accuracy of the graph.

Q: How do you troubleshoot common issues with s-t graphs and a-s graphs?

A: To troubleshoot common issues with s-t graphs and a-s graphs, you need to:

Identify the problem and the relevant information
Check the accuracy of the graph
Use the correct units
Consider the limitations of the graph.

Q: What are some common applications of s-t graphs and a-s graphs in real-world scenarios?

A: Some common applications of s-t graphs and a-s graphs in real-world scenarios include:

Describing the motion of objects in physics and engineering
Predicting the behavior of objects in different situations
Designing and optimizing systems
Analyzing data and making predictions about future behavior.

Q: How do you use s-t graphs and a-s graphs to solve problems in real-world scenarios?

A: To use s-t graphs and a-s graphs to solve problems in real-world scenarios, you need to:

Identify the problem and the relevant information
Determine the type of motion (e.g. constant speed, acceleration, etc.)
Use the s-t graph to determine the distance traveled by the object
Use the a-s graph to determine the acceleration of the object
Combine this information to determine the time required for the object to travel a certain distance.

Q: What are some common mistakes to avoid when working with s-t graphs and a-s graphs in real-world scenarios?

A: Some common mistakes to avoid when working with s-t graphs and a-s graphs in real-world scenarios include:

Failing to take into account the initial conditions of the problem
Failing to consider the limitations of the graph
Failing to use the correct units
Failing to check the accuracy of the graph.

Q: How do you troubleshoot common issues with s-t graphs and a-s graphs in real-world scenarios?

A: To troubleshoot common issues with s-t graphs and a-s graphs in real-world scenarios, you need to:

Identify the problem and the relevant information
Check the accuracy of the graph
Use the correct units
Consider the limitations of the graph.

Q: What are some common tools and software used to create s-t graphs and a-s graphs in real-world scenarios?

A: Some common tools and software used to create s-t graphs and a-s graphs in real-world scenarios include:

Graphing calculators
Computer software (e.g. MATLAB, Python