Distance And Displacement Worksheet With Answers

Distance and Displacement Worksheet: A Comprehensive Guide with Answers

Understanding distance and displacement is fundamental to grasping the concepts of motion in physics. While seemingly similar, these two terms represent distinct aspects of an object's movement. This comprehensive worksheet will help you solidify your understanding, providing clear explanations, practice problems, and detailed solutions.

What is Distance?

Distance is a scalar quantity, meaning it only has magnitude (size) and no direction. It measures the total length of the path traveled by an object, regardless of its starting and ending points. Think of it as the odometer reading in your car – it keeps track of the total distance driven, regardless of the route taken.

Key Characteristics of Distance:

• Scalar: Only magnitude (size).
• Total path length: Accounts for every twist, turn, and detour.
• Always positive: Distance can never be negative.

Example: If you walk 5 meters north, then 3 meters east, the total distance traveled is 5 + 3 = 8 meters.

What is Displacement?

Displacement, on the other hand, is a vector quantity, possessing both magnitude and direction. It measures the shortest distance between an object's starting point and its ending point, taking into account the direction of movement. Think of it as the "as the crow flies" distance.

Key Characteristics of Displacement:

• Vector: Has both magnitude and direction (e.g., 10 meters north).
• Shortest distance: The straight line connecting the start and end points.
• Can be positive, negative, or zero: The sign indicates direction (e.g., positive for forward, negative for backward).

Example: Using the same example above, if you walk 5 meters north, then 3 meters east, your displacement is the straight-line distance from your starting point to your ending point. This can be calculated using the Pythagorean theorem: √(5² + 3²) ≈ 5.83 meters (with a direction specified, e.g., northeast).

Worksheet: Distance and Displacement Problems

Let's put your understanding to the test with a series of problems. Remember to clearly identify both the distance and displacement for each scenario.

Problem 1: A car travels 10 km east, then 5 km south, and finally 2 km west. What is the total distance traveled? What is the displacement?

Problem 2: A jogger runs 200 meters north, then 150 meters south, and finally 50 meters north. What is the total distance traveled? What is the final displacement?

Problem 3: A bird flies 50 meters directly east, then 30 meters directly north. What is the total distance traveled? What is the magnitude of the bird's displacement? What is the direction of the displacement (relative to east)?

Problem 4: A student walks 20 meters to the library, then 15 meters back towards their dorm. What is the distance traveled? What is the displacement from their starting point?

Problem 5: A boat sails 10 km due north, then 20 km due east, and finally 15 km due south. Find the total distance and the magnitude of the displacement.

Problem 6 (Challenging): A hiker walks 3 km north, 4 km east, 2 km south, and 1 km west. Calculate the total distance and the magnitude of the displacement.

Solutions to Distance and Displacement Problems

Let's go through the solutions, step-by-step. Remember, distance is simply the sum of all the distances traveled. Displacement requires considering both the magnitude and direction using vector addition (or, in simpler cases, the Pythagorean theorem).

Solution 1:

• Distance: 10 km + 5 km + 2 km = 17 km
• Displacement: We can use vectors here. The eastward movement (10 km) partially cancels the westward movement (2 km), resulting in a net eastward movement of 8 km. The southward movement is 5 km. The displacement is the hypotenuse of a right-angled triangle with sides 8 km and 5 km. Using the Pythagorean theorem: √(8² + 5²) ≈ 9.43 km (southeast).

Solution 2:

• Distance: 200 m + 150 m + 50 m = 400 m
• Displacement: The northward movement (200 m) partially cancels the southward movement (150 m), leaving a net northward movement of 50 m. The final displacement is 50 m north.

Solution 3:

• Distance: 50 m + 30 m = 80 m
• Displacement (Magnitude): Using the Pythagorean theorem: √(50² + 30²) ≈ 58.3 m
• Displacement (Direction): We can use trigonometry to find the angle. The angle (θ) relative to east is given by: tan(θ) = 30/50, therefore θ = arctan(30/50) ≈ 31°. The displacement is approximately 58.3 m at 31° north of east.

Solution 4:

• Distance: 20 m + 15 m = 35 m
• Displacement: The student walked 20m in one direction and 15m in the opposite direction. Therefore, the displacement is 20m - 15m = 5m in the direction of the library.

Solution 5:

• Distance: 10 km + 20 km + 15 km = 45 km
• Displacement: The northward movement (10 km) partially cancels the southward movement (15 km), resulting in a net southward movement of 5 km. The eastward movement is 20 km. Using the Pythagorean theorem: √(20² + 5²) ≈ 20.6 km (southeast).

Solution 6:

• Distance: 3 km + 4 km + 2 km + 1 km = 10 km
• Displacement: The northward movement (3 km) partially cancels the southward movement (2 km), resulting in a net northward movement of 1 km. The eastward movement (4 km) partially cancels the westward movement (1 km), resulting in a net eastward movement of 3 km. Using the Pythagorean theorem: √(1² + 3²) ≈ 3.16 km (northeast).

Further Practice and Advanced Concepts

This worksheet provides a solid foundation in understanding distance and displacement. To further enhance your understanding, consider exploring these advanced concepts:

• Vector Addition: Learn different methods for adding vectors, such as graphical methods (tip-to-tail) and component methods.
• Relative Motion: Analyze motion from different frames of reference (e.g., a person walking on a moving train).
• Two-dimensional motion: Extend your understanding to include motion in two dimensions, considering both x and y components of displacement and velocity.
• Calculus and Motion: Use calculus to determine displacement as a function of time and velocity from acceleration.

By working through these problems and exploring these advanced topics, you'll develop a comprehensive understanding of distance and displacement, essential for success in physics and related fields. Remember to always pay close attention to both magnitude and direction when dealing with vector quantities like displacement. Consistent practice and conceptual understanding are key to mastering this crucial aspect of physics.