A dolphin wants to swim directly back to its home bay, which is 0.800 km due west. It can swim at a speed of 3.58 m/s relative to the water, but a uniform water current flows with speed 2.95 m/s in the southeast direction.
draw the diagram. t is time swimming
2.95*t*sin135S + 2.95tcos(135)E+3.58tSinTheta N+3.56tCosTheta W = 800W
Now looking at NS only
2.95*t*sin135S +3.58tSinTheta N=0
t divides out, so
Theta= arcsin 2.95/3.58= 180+55.5 deg, that is the direction he has to swim. Now, how long? Look in the EW direction.
2.95tcos(135)E+ 3.56tCosTheta W = 800W
time=800/(2.95 cos(135) - 3.56 Cos55.5)=11317 sec which is a long time.
check my math.
an inclimed plane rises to a height of 2 m over a distance of 6m calculate:the angle of the slope?
To find the relative velocity of the dolphin with respect to the ground, we need to consider the vector addition of the dolphin's swimming speed and the water current velocity.
Step 1: Split the velocities into their x and y components.
The dolphin's swimming speed:
- V_dolphin_x = 3.58 m/s (since it is swimming due west, the x-component is negative)
- V_dolphin_y = 0 m/s (there is no north/south component for swimming due west)
The water current velocity:
- V_current_x = 2.95 m/s (since it is flowing in the southeast direction, the x-component is positive)
- V_current_y = -2.95 m/s (since it is flowing in the southeast direction, the y-component is negative)
Step 2: Add the x and y components separately.
V_relative_x = V_dolphin_x + V_current_x
= -3.58 m/s + 2.95 m/s
= -0.63 m/s
V_relative_y = V_dolphin_y + V_current_y
= 0 m/s + (-2.95 m/s)
= -2.95 m/s
Step 3: Calculate the magnitude and direction of the relative velocity.
The magnitude of the relative velocity can be calculated using the Pythagorean theorem:
V_relative = sqrt(V_relative_x^2 + V_relative_y^2)
= sqrt((-0.63 m/s)^2 + (-2.95 m/s)^2)
= sqrt(0.3969 m^2/s^2 + 8.7025 m^2/s^2)
= sqrt(9.0994 m^2/s^2)
≈ 3.02 m/s (rounded to 3 significant figures)
The direction of the relative velocity can be found using the inverse tangent function:
θ = tan^(-1)(V_relative_y / V_relative_x)
= tan^(-1)(-2.95 m/s / -0.63 m/s)
≈ 77.9° (rounded to 3 significant figures)
Therefore, the dolphin's relative velocity with respect to the ground is approximately 3.02 m/s in the direction of 77.9°, or 77.9° west of south.
To determine how long it will take for the dolphin to swim back to its home bay, we need to consider the combination of its swimming speed and the water current.
Step 1: Calculate the effective velocity of the dolphin
The effective velocity of the dolphin is the vector sum of its swimming speed and the water current. The swimming speed is given as 3.58 m/s to the west, and the water current has a velocity of 2.95 m/s in the southeast direction.
To combine these velocities, we can break them down into their horizontal and vertical components. The swimming speed of 3.58 m/s to the west can be represented as (-3.58, 0) m/s, and the water current with a velocity of 2.95 m/s in the southeast direction can be represented as (2.95 * cos(45°), -2.95 * sin(45°)) m/s.
Note: We use the cosine and sine functions with angle 45° because the southeast direction is halfway between south and east.
Now we can calculate the effective velocity:
Effective velocity = Dolphin's swimming speed + Water current's velocity
= (-3.58, 0) m/s + (2.95 * cos(45°), -2.95 * sin(45°)) m/s
= (-3.58 + 2.95 * cos(45°), -2.95 * sin(45°)) m/s
Step 2: Calculate the time taken to swim back to the home bay
The time taken to swim back to the home bay can be calculated using the formula:
Time = Distance / Speed
Here, the distance is given as 0.800 km, which can be converted to meters: 0.800 km * 1000 m/km = 800 m.
To calculate the speed, we use the magnitude of the effective velocity:
Speed = magnitude of the effective velocity
Magnitude of the effective velocity = sqrt((vx)^2 + (vy)^2)
= sqrt((-3.58 + 2.95 * cos(45°))^2 + (-2.95 * sin(45°))^2)
Finally, we can calculate the time taken:
Time = Distance / Speed
= 800 m / Magnitude of the effective velocity
Once you substitute the values and calculate the expression, you will have the time it takes for the dolphin to swim back to its home bay.