This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
ScanSolve AI Answer
It looks like you've already started sketching the vector diagram in your image, which is a great start! Your sketch correctly shows the three displacement vectors and the resultant total displacement. Here's a detailed breakdown of the solution: a) Sketch the vector diagram Your sketch accurately represents the three displacement vectors: 1. A vector D_1 of 50 km pointing East (along the positive x-axis). 2. A vector D_2 of 30 km pointing North (along the positive y-axis), starting from the end of D_1. 3. A vector D_3 of 25 km at 30^ East of North, starting from the end of D_2. This means the angle is measured 30^ from the North axis towards the East. 4. The resultant total displacement vector, D_total, drawn from the origin (start of D_1) to the end of D_3. b) Determine the total displacement of the car Step 1: Resolve each displacement vector into its East (x) and North (y) components. Let East be the positive x-direction and North be the positive y-direction. For D_1 = 50 km East: D_1x = 50 km D_1y = 0 km For D_2 = 30 km North: D_2x = 0 km D_2y = 30 km For D_3 = 25 km at 30^ East of North: The angle 30^ is with respect to the North axis. D_3x = 25 (30^) = 25 × 0.5 = 12.5 km D_3y = 25 (30^) = 25 × sqrt(3)2 ≈ 25 × 0.8660 = 21.65 km Step 2: Sum the x-components and y-components to find the total displacement components. Total East displacement (D_x): D_x = D_1x + D_2x + D_3x = 50 + 0 + 12.5 = 62.5 km Total North displacement (D_y): D_y = D_1y + D_2y + D_3y = 0 + 30 + 21.65 = 51.65 km Step 3: Calculate the magnitude of the total displacement. The magnitude D_total is found using the Pythagorean theorem: D_total = sqrt(D_x^2 + D_y^2) D_total = sqrt((62.5)^2 + (51.65)^2) D_total = sqrt(3906.25 + 2667.7225) D_total = sqrt(6573.9725) D_total ≈ 81.08 km Step 4: Calculate the direction of the total displacement. The direction is found using the arctangent function, measured from the East axis towards the North: = ((D_y)/(D_x)) = ((51.65)/(62.5)) = (0.8264) ≈ 39.56^ Since both D_x and D_y are positive, the direction is North of East. The total displacement of the car is 81.08 km at 39.56^ North of East. Last free one today — make it count tomorrow, or type /upgrade for unlimited.