A girl throws a water-filled balloon at an angle of 50.0∘^\circ above the horizontal with a speed of 12.0m/s{\rm m}/{\rm s} . The horizontal component of the balloon's velocity is directed toward a car that is approaching the girl at a constant speed of 8.00m/s{\rm m}/{\rm s} .
To determine what happens when the water-filled balloon is thrown, we need to analyze the motion of the balloon both horizontally and vertically.
Let's break down the given information:
1. The angle of 50.0∘ is measured above the horizontal. This means that the angle with respect to the ground is (90∘ - 50.0∘) = 40.0∘.
2. The speed of the balloon is given as 12.0 m/s. This is the magnitude of the initial velocity of the balloon.
3. The car is moving towards the girl at a constant speed of 8.00 m/s.
Now let's determine the horizontal and vertical components of the balloon's velocity:
The horizontal component of the velocity is given by:
Vx = V * cos(θ)
Where V is the initial speed of the balloon and θ is the angle above the horizontal.
Vx = 12.0 m/s * cos(40.0∘)
Vx = 12.0 m/s * 0.766
Vx = 9.1904 m/s (rounded to four decimal places)
The vertical component of the velocity is given by:
Vy = V * sin(θ)
Vy = 12.0 m/s * sin(40.0∘)
Vy = 12.0 m/s * 0.6428
Vy = 7.714 m/s (rounded to three decimal places)
Now, let's determine the time it takes for the balloon to reach its highest point. At the highest point, the vertical velocity becomes zero.
Using the equation:
Vy = Vy0 + a * t
where Vy is the vertical component of velocity, Vy0 is the initial vertical component of velocity, a is the acceleration due to gravity (-9.8 m/s^2) and t is the time.
0 = 7.714 m/s - 9.8 m/s^2 * t
Solving for t:
t = 7.714 m/s / (9.8 m/s^2)
t ≈ 0.787 s (rounded to three decimal places)
Now, let's determine the maximum height the balloon reaches using the formula:
Δy = Vy0 * t + 0.5 * a * t^2
Where Δy is the displacement in the vertical direction, Vy0 is the initial vertical component of velocity, t is the time, and a is the acceleration due to gravity (-9.8 m/s^2).
Δy = 7.714 m/s * 0.787 s + 0.5 * (-9.8 m/s^2) * (0.787 s)^2
Δy ≈ 3.036 m (rounded to three decimal places)
The balloon reaches a maximum height of approximately 3.036 meters.
Now let's analyze the horizontal motion of the balloon. The horizontal velocity of the balloon remains constant throughout its motion because there is no horizontal force acting on it.
The horizontal distance traveled by the balloon can be determined using the equation:
d = Vx * t
where d is the horizontal distance traveled, Vx is the horizontal component of velocity, and t is the time taken.
d = 9.1904 m/s * 0.787 s
d ≈ 7.236 m (rounded to three decimal places)
Therefore, the horizontal distance traveled by the balloon is approximately 7.236 meters.
In summary, when the girl throws the water-filled balloon at an angle of 50.0∘ above the horizontal with a speed of 12.0 m/s, the balloon reaches a maximum height of approximately 3.036 meters and travels a horizontal distance of approximately 7.236 meters. Meanwhile, the car is approaching the girl at a constant speed of 8.00 m/s.