A)You are in a hot air balloon rising from the ground at a constant velocity of 2.50 m/s upward. To celebrate the takeoff, you open a bottle of champagne, expelling the cork with a horizontal velocity of 5.70 m/s relative to the balloon. When opened, the bottle is 6.40 m above the ground.
What is the initial speed of the cork, as seen by your friend on the ground?
B)What is the initial direction of the cork as seen by your friend? Give your answer as an angle relative to the horizontal.
C)Determine the maximum height of the cork above the ground.
D)How long does the cork remain in the air?
A) To find the initial speed of the cork as seen by your friend on the ground, we need to combine the vertical velocity of the hot air balloon with the horizontal velocity of the cork.
The vertical velocity of the hot air balloon is 2.50 m/s upward.
The horizontal velocity of the cork is given as 5.70 m/s.
The initial speed of the cork can be found by using the Pythagorean theorem:
Initial speed = √((vertical velocity)^2 + (horizontal velocity)^2)
Initial speed = √((2.50 m/s)^2 + (5.70 m/s)^2)
Initial speed = √(6.25 m^2/s^2 + 32.49 m^2/s^2)
Initial speed = √38.74 m^2/s^2
Initial speed ≈ 6.23 m/s
Therefore, the initial speed of the cork, as seen by your friend on the ground, is approximately 6.23 m/s.
B) To find the initial direction of the cork as seen by your friend on the ground, we need to calculate the angle relative to the horizontal.
The horizontal velocity of the cork is given as 5.70 m/s.
The vertical velocity of the hot air balloon is 2.50 m/s upward.
The initial direction of the cork can be found by using trigonometry:
tan(angle) = (vertical velocity) / (horizontal velocity)
tan(angle) = (2.50 m/s) / (5.70 m/s)
angle ≈ 24.88 degrees
Therefore, the initial direction of the cork, as seen by your friend on the ground, is approximately 24.88 degrees relative to the horizontal.
C) The maximum height of the cork above the ground can be determined using the equations of motion.
The vertical velocity of the hot air balloon is 2.50 m/s upward.
The acceleration due to gravity is 9.8 m/s^2, acting downward.
Using the equation for vertical displacement:
vertical displacement = (initial vertical velocity)(time) + (1/2)(acceleration due to gravity)(time)^2
At the maximum height, the vertical displacement is 6.40 m and the vertical velocity is 0.
0 = (2.50 m/s)(time) + (1/2)(-9.8 m/s^2)(time)^2
Simplifying the equation:
0 = 2.50t - 4.9t^2
Rearranging the equation to find the time:
4.9t^2 - 2.50t = 0
Applying the zero product property:
t(4.9t - 2.50) = 0
This equation has two solutions: t = 0 (at the start) or 4.9t - 2.50 = 0.
Solving for t, we get:
4.9t - 2.50 = 0
4.9t = 2.50
t ≈ 0.51 s
Therefore, it takes approximately 0.51 seconds for the cork to reach its maximum height.
D) The total time the cork remains in the air is twice the time it takes to reach the maximum height, since it will take the same amount of time to fall back down.
Total time = 2 * (0.51 s)
Total time ≈ 1.02 seconds
Therefore, the cork remains in the air for approximately 1.02 seconds.
A) To find the initial speed of the cork as seen by your friend on the ground, we need to consider the relative velocities. The vertical component of the velocity of the cork will be the same as the ascending balloon, as there is no vertical acceleration. However, the horizontal component of the velocity of the cork will be the sum of the horizontal velocity relative to the balloon and the velocity of the balloon relative to the ground.
Given:
Vertical velocity of the balloon (v_b) = 2.50 m/s upward
Horizontal velocity of the cork relative to the balloon (v_c_rel) = 5.70 m/s
To find the initial speed of the cork as seen by your friend on the ground, we can use Pythagoras' theorem:
Initial speed of the cork (v_c) = √(v_b^2 + v_c_rel^2)
= √(2.50^2 + 5.70^2)
≈ 6.317 m/s
Therefore, the initial speed of the cork, as seen by your friend on the ground, is approximately 6.317 m/s.
B) To find the initial direction of the cork, we need to determine the angle it makes with the horizontal. We can use trigonometry to find this angle.
Given:
Vertical velocity of the balloon (v_b) = 2.50 m/s upward
Horizontal velocity of the cork relative to the balloon (v_c_rel) = 5.70 m/s
To find the initial direction of the cork, we can use the tangent function:
Angle (θ) = tan^(-1)(v_b / v_c_rel)
= tan^(-1)(2.50 / 5.70)
≈ 23.86 degrees
Therefore, the initial direction of the cork, as seen by your friend, is approximately 23.86 degrees relative to the horizontal.
C) To determine the maximum height of the cork above the ground, we need to know the time it takes for the cork to reach its peak. Since the cork is only affected by vertical velocity, we can use the equation of motion for vertical motion:
Displacement (s) = (v_i × t) + (1/2 × a × t^2)
Since the cork begins with an upwards velocity and ends with a downwards velocity (due to gravity), its initial vertical velocity will be the same magnitude but opposite in direction to the vertical velocity of the balloon, i.e., v_i = -v_b = -2.50 m/s. The vertical acceleration (a) is the acceleration due to gravity and is approximately 9.8 m/s². The displacement (s) will be the maximum height reached by the cork (H). We can assume the total time of flight is t_f. At the maximum height, the final vertical velocity will be 0.
Using the equation:
0 = (-2.50 m/s) + (1/2 × 9.8 m/s² × t_f)
5t_f = 2.50
t_f = 0.50 seconds
Now, using this time, we can find the maximum height using the equation of motion:
H = (v_i × t_f) + (1/2 × a × t_f^2)
= (-2.50 m/s × 0.50 s) + (1/2 × 9.8 m/s² × (0.50 s)^2)
≈ -1.25 m + 1.225 m
≈ 0.975 m
Therefore, the maximum height of the cork above the ground is approximately 0.975 meters.
D) The total time of flight for the cork can be found by doubling the time it took to reach the maximum height, since the ascent and descent phases are symmetrical.
Total time of flight (t_total) = 2 × t_f
= 2 × 0.50 s
= 1.00 s
Therefore, the cork remains in the air for approximately 1.00 second.
initial speed is √(2.5^2 + 5.7^2)
The direction is problematical, depending on where the friend stands. If directly below the balloon, its direction appears horizontal. If sufficiently far to the side, then it will appears as being at an angle θ such that tanθ = 2.5/5.7
h = 6.4 + 2.5t - 4.9t^2
find the vertex of that parabola for max height
solve for h=0 to find the time in the air.