Vector Addition - Free Printable
Educational worksheet: Vector Addition. Download and print for classroom or home learning activities.
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Step-by-step solution for: Vector Addition
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Show Answer Key & Explanations
Step-by-step solution for: Vector Addition
To solve the problem, we need to analyze each scenario by considering the vectors and their directions. Let's break it down step by step.
- Two forces of 10 N are acting at an angle to each other.
- One force is directed upward (10 N).
- The other force is directed diagonally downward (10 N).
#### Step-by-Step Analysis:
1. Identify the forces: Both forces have a magnitude of 10 N.
2. Determine the angle: The forces are not aligned in the same direction or opposite directions. They form an angle with each other.
3. Resultant force calculation: Since the exact angle between the forces is not specified, we cannot calculate the exact magnitude of the resultant force without additional information. However, we can say that the resultant force will be less than the sum of the magnitudes (20 N) but more than the difference (0 N).
- If the angle is 90°, the resultant force would be:
\[
R = \sqrt{F_1^2 + F_2^2} = \sqrt{10^2 + 10^2} = \sqrt{200} = 10\sqrt{2} \approx 14.14 \, \text{N}
\]
- If the angle is 180°, the resultant force would be:
\[
R = |F_1 - F_2| = |10 - 10| = 0 \, \text{N}
\]
- For other angles, the resultant force would be somewhere between 0 N and 20 N.
Since the exact angle is not given, we cannot provide a precise numerical answer. However, we can conclude that the resultant force is not zero and depends on the angle.
- Two forces of 10 N are acting in opposite directions.
- One force is directed upward (10 N).
- The other force is directed downward (10 N).
#### Step-by-Step Analysis:
1. Identify the forces: Both forces have a magnitude of 10 N.
2. Determine the direction: The forces are acting in exactly opposite directions.
3. Resultant force calculation: When two equal forces act in opposite directions, they cancel each other out.
\[
R = |F_1 - F_2| = |10 - 10| = 0 \, \text{N}
\]
Therefore, the resultant force is:
\[
\boxed{0 \, \text{N}}
\]
- One force of 10 N is directed upward.
- Another force of 5 N is directed horizontally to the right.
#### Step-by-Step Analysis:
1. Identify the forces: One force is 10 N upward, and the other is 5 N to the right.
2. Determine the direction: The forces are perpendicular to each other.
3. Resultant force calculation: Since the forces are perpendicular, we can use the Pythagorean theorem to find the magnitude of the resultant force.
\[
R = \sqrt{F_1^2 + F_2^2} = \sqrt{10^2 + 5^2} = \sqrt{100 + 25} = \sqrt{125} = 5\sqrt{5} \approx 11.18 \, \text{N}
\]
The direction of the resultant force can be found using trigonometry, but the magnitude is:
\[
\boxed{5\sqrt{5} \, \text{N}}
\]
1. Scenario 1: Depends on the angle (cannot be determined precisely without the angle).
2. Scenario 2: \(\boxed{0 \, \text{N}}\)
3. Scenario 3: \(\boxed{5\sqrt{5} \, \text{N}}\)
Scenario 1:
- Two forces of 10 N are acting at an angle to each other.
- One force is directed upward (10 N).
- The other force is directed diagonally downward (10 N).
#### Step-by-Step Analysis:
1. Identify the forces: Both forces have a magnitude of 10 N.
2. Determine the angle: The forces are not aligned in the same direction or opposite directions. They form an angle with each other.
3. Resultant force calculation: Since the exact angle between the forces is not specified, we cannot calculate the exact magnitude of the resultant force without additional information. However, we can say that the resultant force will be less than the sum of the magnitudes (20 N) but more than the difference (0 N).
- If the angle is 90°, the resultant force would be:
\[
R = \sqrt{F_1^2 + F_2^2} = \sqrt{10^2 + 10^2} = \sqrt{200} = 10\sqrt{2} \approx 14.14 \, \text{N}
\]
- If the angle is 180°, the resultant force would be:
\[
R = |F_1 - F_2| = |10 - 10| = 0 \, \text{N}
\]
- For other angles, the resultant force would be somewhere between 0 N and 20 N.
Since the exact angle is not given, we cannot provide a precise numerical answer. However, we can conclude that the resultant force is not zero and depends on the angle.
Scenario 2:
- Two forces of 10 N are acting in opposite directions.
- One force is directed upward (10 N).
- The other force is directed downward (10 N).
#### Step-by-Step Analysis:
1. Identify the forces: Both forces have a magnitude of 10 N.
2. Determine the direction: The forces are acting in exactly opposite directions.
3. Resultant force calculation: When two equal forces act in opposite directions, they cancel each other out.
\[
R = |F_1 - F_2| = |10 - 10| = 0 \, \text{N}
\]
Therefore, the resultant force is:
\[
\boxed{0 \, \text{N}}
\]
Scenario 3:
- One force of 10 N is directed upward.
- Another force of 5 N is directed horizontally to the right.
#### Step-by-Step Analysis:
1. Identify the forces: One force is 10 N upward, and the other is 5 N to the right.
2. Determine the direction: The forces are perpendicular to each other.
3. Resultant force calculation: Since the forces are perpendicular, we can use the Pythagorean theorem to find the magnitude of the resultant force.
\[
R = \sqrt{F_1^2 + F_2^2} = \sqrt{10^2 + 5^2} = \sqrt{100 + 25} = \sqrt{125} = 5\sqrt{5} \approx 11.18 \, \text{N}
\]
The direction of the resultant force can be found using trigonometry, but the magnitude is:
\[
\boxed{5\sqrt{5} \, \text{N}}
\]
Final Answers:
1. Scenario 1: Depends on the angle (cannot be determined precisely without the angle).
2. Scenario 2: \(\boxed{0 \, \text{N}}\)
3. Scenario 3: \(\boxed{5\sqrt{5} \, \text{N}}\)
Parent Tip: Review the logic above to help your child master the concept of vector addition worksheet with answers.
