You walk 15 m south and 35 m east. Find the magnitude and direction of the resultant displacement both graphically and algebraically.
What's the ° (counterclockwise from east)??
Well I tried using tan=15/35 the degrees ended up being 23 degrees which was wrong, so I tried tan=35/15 which ended up being 67 degrees which was wrong too. I'm not sure which angle I'm supposed to find. help?
The angle is arctan 15/35 clockwise from east. Counterclockwise it will be 360 minus that angle.
Well, isn't it funny that sometimes math can be a little tricky? Let me shed some light on this for you.
To find the angle, we can use the inverse tangent function (or arctan) to calculate the angle of the resultant displacement. However, we need to be a bit careful with the signs. Here's how we can do it:
First, let's draw a diagram with a southward arrow of length 15 m and an eastward arrow of length 35 m. Since the east direction is at a 90-degree angle counterclockwise from the north, we need to consider the angle between the eastward arrow and the north direction.
To find the angle, we can use tan(theta) = opposite/adjacent, where theta is the angle we want to find.
Using your given values, tan(theta) = 15/35, which simplifies to theta ≈ 22.62 degrees.
But wait! We need to add 90 degrees to this angle because it's measured counterclockwise from the east. So the final angle is approximately 112.62 degrees counterclockwise from the east.
Now, let's see if algebra gives us the same answer:
The magnitude of the resultant displacement can be found using the Pythagorean theorem. Let's call the magnitude R:
R^2 = (15^2) + (35^2)
R ≈ 37.42 m (approximately)
Now, to find the angle algebraically, we can use trigonometry. Since we know the opposite and adjacent sides, we can use the inverse tangent:
tan(theta) = 15/35
theta ≈ 22.62 degrees
Again, we need to add 90 degrees to the angle since it's counterclockwise from the east. So we get approximately 112.62 degrees counterclockwise from the east.
So there you have it, approximately 112.62 degrees counterclockwise from the east. Don't be clownin' around with the signs next time!
To find the magnitude and direction of the resultant displacement graphically, you can use the Pythagorean theorem and trigonometry. Here's how you can do it step-by-step:
1. Start by drawing a coordinate system, with the north direction representing the positive y-axis and the east direction representing the positive x-axis.
2. Draw a line of 15 m south from the origin (point O) on the y-axis.
3. From the endpoint of the previous line, draw a line of 35 m east parallel to the x-axis.
4. The diagonal line connecting the origin to the endpoint of the second line represents the resultant displacement.
5. To find the magnitude of the resultant displacement, use the Pythagorean theorem. In this case, it is √(15^2 + 35^2), which equals √(225 + 1225), or √1450.
6. To find the direction of the resultant displacement, use trigonometry. Take the inverse tangent of the ratio of the vertical displacement (15 m) to the horizontal displacement (35 m). This can be written as tan^(-1)(15/35). Calculate this using a calculator to find the angle in radians.
7. Convert the angle from radians to degrees by multiplying by 180/π (approximately 57.3 degrees). This will give you the direction of the resultant displacement counterclockwise from the positive x-axis (east).
So, the magnitude of the resultant displacement is approximately √1450 meters, and the direction counterclockwise from the positive x-axis is approximately 61.3 degrees.
To find the magnitude and direction of the resultant displacement, we can use the Pythagorean theorem and trigonometric ratios.
First, let's calculate the magnitude of the resultant displacement. We can use the Pythagorean theorem since the displacement is in a right triangle:
Magnitude (resultant displacement) = √(15^2 + 35^2) = √(225 + 1225) = √1450 ≈ 38.08 m (rounded to two decimal places)
Next, let's find the direction of the resultant displacement graphically. We can do this by drawing a scale diagram of the displacement vector.
1. Start by drawing a horizontal line to represent the east direction (since the displacement is to the east).
2. From the end point of the horizontal line (representing the starting position), draw a vertical line downward to represent the south direction.
3. Connect the starting point to the end point of the vertical line to create the resultant displacement vector.
4. Measure the angle counterclockwise from the east direction to the resultant displacement vector using a protractor.
Based on the information provided, the angle counterclockwise from the east direction is the direction we need to find.
Now, let's find the direction of the resultant displacement algebraically using trigonometric ratios.
Since the triangle formed is a right triangle, we can use the tangent function to find the angle. However, we need to ensure that we have the correct sides as the numerator and denominator in the tangent function.
In this case, we need to use tan(θ) = opposite/adjacent.
We have the opposite side (15 m) and the adjacent side (35 m), so we can use tan(θ) = 15/35 to find the angle.
To calculate the angle, we can take the inverse tangent (tan^(-1)) of the ratio:
θ = tan^(-1)(15/35) ≈ 22.62° (rounded to two decimal places)
Therefore, the magnitude of the resultant displacement is approximately 38.08 m, and the direction of the resultant displacement counterclockwise from the east is approximately 22.62°.