A dolphin jumps with an initial velocity of 15 m/s at an angle of 45 degrees above the horizontal. The dolphin passes through the center of a hoop before returning to the water. The dolphin is moving horizontally when it goes through the hoop.
a) What are the x and y components of the initial velocity?
b) What are the acceleration in horizontal and vertical direction?
c) How high above the water is the center of the hoop?
Answers
Explanation:
a) u=15/√2î + 15/√2j
b) a=0î - 10j
c)H max.=5.625m above the ground
Answer:
a) The x and y components of the initial velocity are both 10.6 m/s.
b) 0 m/s^2, 9.8 m/s^2
c) The center of the hoop is 5.45 meters above the water.
Explanation:
a) To find the x and y components of the initial velocity, we can use trigonometry. Following are the horizontal and vertical components of the initial velocity:
Vx = V * cos(theta)
Vy = V * sin(theta)
where V is the magnitude of the initial velocity (15 m/s) and theta is the angle of the initial velocity above the horizontal (45 degrees).
Substituting the given values, we get:
Vx = 15 * cos(45) = 10.6 m/s
Vy = 15 * sin(45) = 10.6 m/s
So the x and y components of the initial velocity are both 10.6 m/s.
b) The acceleration in the horizontal direction is zero, since there are no horizontal forces acting on the dolphin once it leaves the water.
The acceleration in the vertical direction is due to gravity and is equal to 9.8 m/s^2, directed downwards.
c) To find the height of the center of the hoop above the water, we can use kinematic equations. At the highest point of its trajectory, the dolphin's vertical velocity will be zero, and we can use this fact to find the time it takes to reach this point:
Vy = V0y + at
0 = 10.6 + (-9.8)t
t = 1.08 seconds
Next, we can use the time and the vertical velocity to find the height of the dolphin above the water at the highest point of its trajectory:
y = V0y * t + (1/2) * a * t^2
y = 10.6 * 1.08 + (1/2) * (-9.8) * (1.08)^2
y = 5.45 meters
So the center of the hoop is 5.45 meters above the water.
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