Kangaroos can easily jump as far as 8.0 m. If a kangaroo makes five such jumps westward and then seven such jumps northward, what angle would the resultant displacement vector make with the northern displacement vector? What is the magnitude of the total displacement?
s12
To find the angle between the resultant displacement vector and the northern displacement vector, we need to first find the individual components of the resultant displacement vector.
Since the kangaroo makes five jumps westward, the horizontal component of the resultant displacement vector will be 5 times the length of each jump:
Horizontal component = 5 * 8.0 m = 40.0 m (west)
Since the kangaroo makes seven jumps northward, the vertical component of the resultant displacement vector will be 7 times the length of each jump:
Vertical component = 7 * 8.0 m = 56.0 m (north)
To find the angle between the resultant displacement vector and the northern displacement vector, we can use the tangent function:
Tan(θ) = Vertical component / Horizontal component
Plugging the values into the formula, we get:
Tan(θ) = 56.0 m / 40.0 m
θ = atan(56.0 m / 40.0 m)
Using a calculator, we find:
θ = 53.1 degrees
So, the angle between the resultant displacement vector and the northern displacement vector is approximately 53.1 degrees.
To find the magnitude of the total displacement, we can use the Pythagorean theorem:
Total displacement = √(Horizontal component^2 + Vertical component^2)
Plugging in our values, we get:
Total displacement = √(40.0 m^2 + 56.0 m^2)
Total displacement = √(1600.0 m^2 + 3136.0 m^2)
Total displacement = √4736.0 m^2
Total displacement ≈ 68.8 m
So, the magnitude of the total displacement is approximately 68.8 m.
To determine the angle made by the resultant displacement vector with the northern displacement vector and the magnitude of the total displacement, we can use vector addition.
Let's assume each jump of the kangaroo is a vector with a magnitude of 8.0 m. We will use Cartesian coordinates to represent the vectors. The westward jump will have a component of -8.0 in the x-direction, and the northward jump will have a component of +8.0 in the y-direction.
Since the kangaroo makes five westward jumps, the total westward displacement vector will be:
(-8.0 m) x 5 = -40.0 m in the x-direction
And since the kangaroo makes seven northward jumps, the total northward displacement vector will be:
(+8.0 m) x 7 = +56.0 m in the y-direction
To find the resultant displacement vector, we need to add the individual displacement vectors. The resultant displacement vector will be the vector that connects the starting point to the final destination, which is the sum of the westward and northward displacement vectors.
So, the resultant displacement vector will be:
(-40.0 m, +56.0 m)
To find the angle made by the resultant displacement vector with the northern displacement vector, we can use trigonometry. The angle can be found using the inverse tangent function:
angle = arctan((vertical component of the resultant displacement vector) / (horizontal component of the resultant displacement vector))
angle = arctan(56.0 m / -40.0 m)
Using a calculator, we find that the angle is approximately -53.13 degrees. Note that the negative sign indicates that the resultant displacement vector is in the opposite direction of the westward displacement vector.
To find the magnitude of the total displacement, we can use the Pythagorean theorem:
magnitude of the total displacement = sqrt((horizontal component of the resultant displacement vector)^2 + (vertical component of the resultant displacement vector)^2)
magnitude of the total displacement = sqrt((-40.0 m)^2 + (56.0 m)^2)
Using a calculator, we find that the magnitude of the total displacement is approximately 69.28 m.
Therefore, the angle made by the resultant displacement vector with the northern displacement vector is approximately -53.13 degrees, and the magnitude of the total displacement is approximately 69.28 m.