1) A fish jumps out of the water with a speed of 6.36 m/s. If the fish is in a stream with water speed = to 1.5 m/s, how high can the fish jump if it leaves the water traveling vertically upwards relative to the earth?
2)A ball is thrown by someone on a train moving at 10 m/s on a horizontal track with a path that they judge to be 60 degrees with the horizontal and to be in line with the track. Someone on the ground nearby observes the ball to rise vertically. How high does it rise?
The horizontal velocity component has nothing to do with how high either object goes. Just solve the vertical problems once you find the vertical speed.
a = -g = -9.8 m/s^2
v = Vo - 9.8 t
h = Vo t - 4.9 t^2
at the top, v = 0 so
t = Vo/9.8 at the top
h = (1/2) Vo^2/9.8 = Vo^2/ 19.6
so for problem 1
the horizontal component of speed = 1.5 = 6.36 cos A where A is the angle it leaves the water at.
so A = cos^-1 (1.5/3.6) = 65.4 degrees
so the vertical component of speed is
Vo = 6.36 sin 65.4 = 5.78 m/s
use that to get h
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for problem 2
call the speed of the ball S
the horizontal component of velocity is:
10 = S cos 60
S =20 m/s
so the vertical component of S is
Vo = 20 sin 60 = 17.3 m/s
so the vertical component of S
To answer these questions, we need to analyze the motion of the objects involved and use some principles of physics.
1) To find how high the fish can jump, we first need to determine its vertical velocity relative to the earth. We can subtract the velocity of the water stream from the fish's jump velocity.
Vertical velocity of the fish relative to the earth = Fish's jump velocity - Water stream velocity
Vertical velocity = 6.36 m/s - 1.5 m/s
Vertical velocity = 4.86 m/s
Now we can use the kinematic equation for vertical motion to find the height the fish can reach, assuming no air resistance:
Final vertical velocity^2 = Initial vertical velocity^2 + 2 * acceleration * height
Since the fish jumps upward, its final velocity is zero (at the highest point of its trajectory). The initial vertical velocity is 4.86 m/s, and the acceleration due to gravity is -9.8 m/s^2 (negative because it opposes the upward motion).
0^2 = 4.86^2 + 2 * (-9.8) * height
Rearranging the equation and solving for height:
2 * (-9.8) * height = -4.86^2
-19.6 * height = -23.62
height = -23.62 / -19.6
height ≈ 1.20 meters
Therefore, the fish can jump approximately 1.20 meters high relative to the Earth's surface.
2) To find how high the ball rises, we can use similar principles.
From the perspective of the person on the train, the ball is thrown with an initial velocity at an angle of 60 degrees with the horizontal. However, from the perspective of the person on the ground nearby, the ball appears to rise vertically. This means that the vertical component of the ball's initial velocity is equal to its overall initial velocity.
The vertical velocity of the ball is given by the initial velocity multiplied by the sine of the launch angle:
Vertical velocity = Initial velocity * sin(angle)
We do not know the initial velocity, but we can use the fact that the ball rises vertically to deduce its vertical velocity.
Using the cosine of the launch angle, we can determine the horizontal component of the initial velocity:
Horizontal velocity = Initial velocity * cos(angle)
Since the horizontal velocity remains constant during the ball's flight, the observer on the ground will see the ball rise vertically.
Now, knowing the horizontal velocity, we can calculate the time it takes for the ball to reach its maximum height using the concept of time of flight:
Time of flight = 2 * (Vertical velocity) / (Acceleration due to gravity)
Assuming the acceleration due to gravity is -9.8 m/s^2, we can calculate the time of flight.
Next, using the time of flight and the vertical velocity, we can calculate the height the ball reaches using the kinematic equation:
Final vertical position = Initial vertical position + (Vertical velocity * Time of flight) + (0.5 * Acceleration due to gravity * (Time of flight)^2)
Since the initial position is zero (ball thrown from the ground), we can simplify the equation to:
Final vertical position = Vertical velocity * Time of flight + (0.5 * Acceleration due to gravity * (Time of flight)^2)
Now, plugging in the values and calculating:
Height = Vertical velocity * Time of flight + (0.5 * Acceleration due to gravity * (Time of flight)^2)
Finally, we will have the height the ball rises from the ground, as observed by the person on the ground.
Note: To answer the question more precisely, the initial velocity of the ball needs to be known.