A dolphin wants to swim directly back to its home bay, which is 0.85 km due west. It can swim at a speed of 4.35 m/s relative to the water, but a uniform water current flows with speed 2.82 m/s in the southeast direction.
(a) What direction should the dolphin head???
(b) How long does it take the dolphin to swim the 0.85-km distance home???
Please helppp
4.35m/s @ 180 Deg. + 2.82m/s @ 315 Deg.
X = 4.35*cos180+2.82*cos315=-2.356 m/s.
Y = 4.35*sin180+2.82*sin315=-1.994 m/s.
Q3.
tanAr = Y/X = -1.994 / -2.356= 0.84636,
Ar = 40 Deg.
A = 180 + 40 = 220 Deg.
a. Direction = 180 - 40 = 140 Deg.
b. d = V*t,
t = d/V = 850m / 4.35m/s = 195.4 s. =
3.26 min.
Vd=4.35m/s @ 180 Deg-2.82m/s @ 315 Deg=
Velocity of dolphin.
X = 4.35*cos180 - 2.82cos315=-6.34 m/s.
Y = 4.35*sin180 - 2.82*sin315=1.994 m/s
Q2.
tanAr = Y/X = 1.994 / -6.34 = -0.31451
Ar = -17.46 Deg. = Reference angle.
A = -17.46 + 180 = 162.5 Deg.
Vd=X/cosA = -6.34 / cos162.5=6.65 m/s
a. Direction = 180 + 17.5 = 197.5 Deg.
b. d = V*t,
t = d/V = 850m / 6.34=134 s.=2.23min.
To determine the direction the dolphin should head, we need to find the resultant velocity vector by adding the velocities of the dolphin and the water current.
Given:
Dolphin's swim speed = 4.35 m/s
Water current speed = 2.82 m/s
We can break down the water current velocity into its eastward and southward components:
Eastward Component: Velocity * cos(45°) = 2.82 m/s * cos(45°) ≈ 1.9975 m/s
Southward Component: Velocity * sin(45°) = 2.82 m/s * sin(45°) ≈ 1.9975 m/s
To find the resultant velocity of the dolphin, we subtract the water current velocity components from the dolphin's swim velocity (due west):
Resultant Eastward Component = Dolphin's swim speed - Eastward Water Current Component
= 4.35 m/s - 1.9975 m/s ≈ 2.3525 m/s (towards the west)
Resultant Southward Component = Southward Water Current Component
= 1.9975 m/s (towards the south)
Thus, the dolphin should head in a direction that is 2.3525 m/s to the west and 1.9975 m/s to the south.
To calculate how long it takes the dolphin to swim the 0.85 km distance home, we can use the formula:
Time = Distance / Speed
Given:
Distance = 0.85 km = 0.85 * 1000 m = 850 m
Dolphin's swim speed = 4.35 m/s
Time = 850 m / 4.35 m/s ≈ 195.40 seconds
Therefore, it takes the dolphin approximately 195.40 seconds (or 3 minutes and 15.4 seconds) to swim the 0.85 km distance home.
To answer this question, we need to break down the dolphin's swimming velocity into its horizontal and vertical components, taking into account the water current.
(a) To determine the direction the dolphin should head, we need to find the resultant velocity that counteracts the effect of the water current. The dolphin wants to swim directly west (which is in the negative x-direction), so it needs to swim with a velocity that cancels out the eastward (positive x-direction) component of the water current.
Let's call the dolphin's horizontal velocity component Vdolphin_x and the water current's horizontal velocity component Vcurrent_x. Since the dolphin wants to swim directly west, Vdolphin_x = -4.35 m/s.
The water current's velocity is given as 2.82 m/s in the southeast direction. Southeast is between south and east directions, which can be broken down into their x and y components. The southeast direction is 45 degrees from both the positive x-direction and the negative y-direction.
To calculate Vcurrent_x, we need to find the horizontal component of the water current's velocity vector. Since the southeast direction is 45 degrees from both the positive x-direction and the negative y-direction, we can use trigonometry to find the horizontal component:
Vcurrent_x = Vcurrent * cos(45°)
= 2.82 m/s * cos(45°)
= 2.82 m/s * 0.707 (approximating cos(45°) to 0.707)
≈ 1.99 m/s
Now, the dolphin needs to swim with a velocity of -4.35 m/s to counteract the effect of the water current (1.99 m/s in the positive x-direction). By subtracting the water current's horizontal velocity component from the dolphin's own horizontal velocity component, we find:
Vdolphin_x = -4.35 m/s - 1.99 m/s
≈ -6.34 m/s
In summary, the dolphin should head in the direction of approximately -6.34 m/s, which corresponds to heading due west.
(b) To find the time it takes the dolphin to swim the 0.85 km distance home, we need to calculate the time it takes to travel this distance using the dolphin's net horizontal velocity (accounting for the water current).
The net horizontal velocity of the dolphin can be calculated by subtracting the water current's horizontal velocity component from the dolphin's horizontal velocity component:
Net horizontal velocity = Vdolphin_x - Vcurrent_x
= -6.34 m/s - 1.99 m/s
≈ -8.33 m/s
Next, we convert the distance of 0.85 km to meters:
Distance = 0.85 km * 1000 m/km
= 850 m
Finally, we can calculate the time using the formula: time = distance / velocity.
Time = distance / net horizontal velocity
= 850 m / -8.33 m/s
≈ -102.01 s (approximated to the nearest second)
Since time cannot be negative, we take the absolute value of the result:
Time ≈ 102.01 s
In summary, it takes the dolphin approximately 102.01 seconds to swim the 0.85 km distance home.