1. A delivery truck travels 21 blocks north, 16 blocks east, and 26 blocks south. What is its final displacement from the origin? Assume the blocks are equal length.
2. Vector V1 is 6.6 units long and points along the negative x axis. Vector V2 is 8.5 units long and points at + 55° to the positive x axis. (a) What are the x and y components of each vector?
(b) Determine the sum V1 + V2 (magnitude and angle).
3. A diver running 2.5 m/s dives out horizontally from the edge of a vertical cliff and 3.0 s later reaches the water below. How high was the cliff and how far from its base did the diver hit the water?
north = 21 - 26 = - 5
east = 16
distance = sart (5^2+16^2)
on x y axis system
tan h = -5/16
h = -17.35
so 17.35 degrees below x axis
which is a compass bearing clockwise from north of
17.35 + 90 = 107.35
about ESE
1. To find the final displacement from the origin, we need to calculate both the vertical and horizontal displacements separately.
The truck traveled 21 blocks north and then 26 blocks south. Since these two movements are in opposite directions, they cancel each other out. So, the vertical displacement is 0 blocks.
The truck also traveled 16 blocks east. Therefore, the horizontal displacement is 16 blocks east.
To find the final displacement, we can use the Pythagorean theorem to calculate the magnitude of the displacement. In this case, it becomes a right triangle with a horizontal displacement of 16 blocks and a vertical displacement of 0 blocks. The magnitude of the displacement is the hypotenuse of the triangle.
Applying the Pythagorean theorem, the magnitude of the displacement is:
displacement = sqrt((horizontal displacement)^2 + (vertical displacement)^2)
= sqrt((16 blocks)^2 + (0 blocks)^2)
= sqrt(256 blocks^2)
= 16 blocks
Therefore, the truck's final displacement from the origin is 16 blocks to the east.
2. (a) To determine the x and y components of each vector, we can use trigonometry.
For Vector V1, since it points along the negative x-axis, its x-component is -6.6 units, and its y-component is 0 units.
For Vector V2, since it makes an angle of +55° with the positive x-axis, we can decompose it into its x and y components using trigonometry. The x-component is given by cos(55°) times the magnitude of V2, which is 8.5 units. Therefore, the x-component of V2 is approximately 8.5 cos(55°) units. The y-component is given by sin(55°) times the magnitude of V2. So, the y-component of V2 is approximately 8.5 sin(55°) units.
(b) To determine the sum V1 + V2, we need to add the corresponding components of the two vectors.
The x-component of V1 + V2 is the sum of the x-components of V1 and V2: -6.6 units + 8.5 cos(55°) units.
The y-component of V1 + V2 is the sum of the y-components of V1 and V2: 0 units + 8.5 sin(55°) units.
To find the magnitude of V1 + V2, we can use the Pythagorean theorem with the x and y components we just calculated. The magnitude is given by:
magnitude = sqrt((x-component)^2 + (y-component)^2)
Finally, to find the angle of V1 + V2, we can use trigonometry. If we let θ be the angle between V1 + V2 and the positive x-axis, we can use the arctan function to find θ:
θ = arctan(y-component / x-component)
1. To find the final displacement, we need to determine the net displacement in both the north-south and east-west directions.
- In the north-south direction, the truck travels 21 blocks north and then 26 blocks south. The net displacement in this direction is 21 - 26 = -5 blocks south.
- In the east-west direction, the truck travels 16 blocks east. Therefore, the net displacement in this direction is 16 blocks east.
To find the final displacement, we can use the Pythagorean theorem. Let's denote the north-south displacement as Δy and the east-west displacement as Δx. Therefore:
Δy = -5 blocks
Δx = 16 blocks
The final displacement (d) from the origin is given by d = √(Δx^2 + Δy^2). Substituting the values:
d = √(16^2 + (-5)^2)
d = √(256 + 25)
d = √281
d ≈ 16.77 blocks
Therefore, the final displacement from the origin is approximately 16.77 blocks.
2. (a) To find the x and y components of each vector, we can use trigonometry. Let's denote the x-component of V1 as V1x, the y-component as V1y, the x-component of V2 as V2x, and the y-component as V2y.
For V1:
V1 = 6.6 units (length)
The vector points along the negative x-axis, so V1x = -6.6 units and V1y = 0 units.
For V2:
V2 = 8.5 units (length)
The vector points at an angle of +55° to the positive x-axis, so we can use trigonometric functions to determine the x and y components.
V2x = V2 * cos(55°) = 8.5 * cos(55°) ≈ 4.40 units
V2y = V2 * sin(55°) = 8.5 * sin(55°) ≈ 7.04 units
Therefore:
V1x = -6.6 units
V1y = 0 units
V2x ≈ 4.40 units
V2y ≈ 7.04 units
(b) To determine the sum V1 + V2, we can add the x and y components separately and then calculate the magnitude and angle of the resultant vector.
To find the x and y components of the sum:
Vx = V1x + V2x = -6.6 + 4.40 = -2.20 units
Vy = V1y + V2y = 0 + 7.04 = 7.04 units
To calculate the magnitude of the sum (V), we can use the Pythagorean theorem:
V = √(Vx^2 + Vy^2) = √((-2.20)^2 + (7.04)^2) ≈ 7.42 units
To find the angle (θ), we can use inverse trigonometric functions:
θ = tan^(-1)(Vy / Vx) = tan^(-1)(7.04 / -2.20) ≈ -73.18°
Therefore, the magnitude of the sum V1 + V2 is approximately 7.42 units, and the angle is approximately -73.18°.
3. To solve this problem, we need to use the kinematic equation in the horizontal direction to find the distance from the base of the cliff and the kinematic equation in the vertical direction to find the height of the cliff. Let's denote the horizontal distance as d and the height as h.
Horizontal motion (in the x-direction):
The diver runs horizontally for 3.0 seconds, so the distance traveled in the x-direction is given by d = v * t, where v is the horizontal velocity and t is the time.
Given that the diver's horizontal velocity is constant at 2.5 m/s, we have d = 2.5 m/s * 3.0 s = 7.5 meters.
Therefore, the diver hits the water 7.5 meters away from the base of the cliff.
Vertical motion (in the y-direction):
The vertical displacement (h) can be determined using the equation h = v0 * t + (1/2) * a * t^2, where v0 is the initial vertical velocity, t is the time, and a is the acceleration in the vertical direction.
The diver dives out horizontally, so the initial vertical velocity (v0) is 0 m/s (since there is no initial vertical velocity).
Given that the diver is in free fall, the acceleration due to gravity (a) is approximately 9.8 m/s^2 (assuming no air resistance).
Substituting the values into the equation, we have h = 0 * 3.0 + (1/2) * 9.8 * (3.0)^2 = 0 + 14.7 = 14.7 meters.
Therefore, the height of the cliff is 14.7 meters.