A - A boy runs 12.7 blocks North, 7.5 blocks Northeast, and 14.4 blocks West. Determine the length of the displacement vector that goes from the starting point to his final position.
B - Determine the direction of the displacement vector. Use counterclockwise as the positive angular direction, between the limits of −180◦ and +180◦ measured from East. Answer in units of ◦.
A. Disp. = 12.7[90o] + 7.5[45o] + 14.4[180o].
Disp. = 12.7i + (5.30+5.30i) - 14.4
Disp. = 18i - 9.1
Disp. = sqrt(9.1^2+18^2) =
B. Tan A = y/x = 18/(-9.1) = -1.97802.
A = -63.18o = 63.18o N. of W. = 116.8o CCW.
A - Well, this boy really went on a wild adventure, didn't he? Let's calculate his displacement. First, we need to break down his movement into its horizontal and vertical components.
North is the positive direction, so he went 12.7 blocks in that direction. But then he went a little off to the Northeast at a 45-degree angle (halfway between North and East). Using some good old Pythagorean theorem magic, we can find that the boy traveled a total of sqrt(12.7^2 + 7.5^2) blocks in that direction.
But then he decided to head west, oh boy! That's never a good idea. He went 14.4 blocks in that direction.
Now, to find the displacement vector, we subtract the distance he went west from the combined distance he went north and northeast.
Displacement = sqrt(12.7^2 + 7.5^2) - 14.4 blocks.
So, the length of the displacement vector is equal to approximately sqrt(183.94) - 14.4 blocks. You can do the math and get the final answer.
B - Now, to calculate the direction of the displacement vector, we need some good old-fashioned trigonometry. We'll use the inverse tangent function to find the angle.
The direction will be equal to the inverse tangent of the vertical displacement divided by the horizontal displacement (opposite/adjacent).
Direction = arctan(7.5 / 12.7) degrees. Calculate that to get your answer!
Hope that helps, and remember, even if the boy's direction was a little chaotic, at least we got some math fun out of it!
To determine the length of the displacement vector, we need to calculate the magnitude of the resultant displacement. We can use the Pythagorean theorem to find this value.
A - To find the length of the displacement vector:
1. Calculate the vertical displacement: 12.7 blocks (North) - 7.5 blocks (Northeast) = 5.2 blocks North.
2. Calculate the horizontal displacement: 14.4 blocks (West).
3. Use the Pythagorean theorem: displacement = √(vertical displacement^2 + horizontal displacement^2).
displacement = √(5.2^2 + 14.4^2) = √(27.04 + 207.36) = √234.4 ≈ 15.29 blocks.
Therefore, the length of the displacement vector is approximately 15.29 blocks.
B - To determine the direction of the displacement vector:
1. Calculate the angle that the displacement vector makes with the horizontal axis. We can use the arctan function to find this angle.
angle = atan(vertical displacement / horizontal displacement)
angle = atan(5.2 / -14.4) ≈ -19.8°
Since we are using counterclockwise as the positive angular direction, we can represent the angle as a positive value. By adding 180° to the negative angle, we can find the direction measured from the East.
2. Correct the angle by adding 180°:
corrected angle = -19.8° + 180° ≈ 160.2°.
Therefore, the direction of the displacement vector is approximately 160.2°.
To solve both questions, we can use vector addition and basic trigonometry.
For question A:
1. Draw a diagram to represent the boy's movements. Assume the starting point as the origin (0,0). The boy's movements can be represented by three vectors: one going 12.7 blocks North, one going 7.5 blocks Northeast, and one going 14.4 blocks West.
2. Convert the Northeast movement vector into its North and West components. Since Northeast is 45 degrees between North and East, we can split the vector into two equal components, one going North and one going West. These components can be found using basic trigonometry. The North component will be 7.5 * sin(45°) and the West component will be 7.5 * cos(45°).
3. Now, add up all the North and West components. The total North component will be the sum of the 12.7 blocks North and the North component of the Northeast vector. The total West component will be the 14.4 blocks West minus the West component of the Northeast vector.
4. Use the Pythagorean theorem to find the length of the displacement vector. The length is the square root of the sum of the squares of the North and West components.
For question B:
1. Use trigonometry to find the angle between the displacement vector and the positive x-axis (East). We can use the arctan function of the ratio of the North and West components to find this angle. However, due to the limited range of -180° to +180°, we need to adjust the angle accordingly.
2. If the angle is greater than 180°, subtract 360° from it. If the angle is less than -180°, add 360° to it. This adjustment will bring the angle within the desired range.
3. The adjusted angle represents the direction of the displacement vector, measured counterclockwise from the positive x-axis (East).
By following these steps, you should be able to determine both the length and direction of the displacement vector.