A water balloon launcher launches a balloon straight up at a flying kite. If the kite is hovering 75.0 meters off the ground, and the balloon hits it on its upward trajectory with a velocity of 12.2m/s,
a) What was the launch velocity of the balloon
total energy at balloon = launch energy
mgh+ 1/2 m (12.2)^2=1/2 m vlaunch^2
solve for vlaunch
To find the launch velocity of the balloon, we need to understand the concept of projectile motion.
Projectile motion is the motion of an object (in this case, the water balloon) that is launched into the air and is subject only to the force of gravity and air resistance (which we'll ignore for simplicity).
The key principle in projectile motion is that the horizontal and vertical motions are independent of each other. Thus, we can solve for the launch velocity of the balloon by considering the vertical motion of the balloon.
In the vertical direction, the balloon starts from the ground (height = 0) and goes up to a height of 75.0 meters. The vertical motion can be described by the kinematic equation:
Δy = v0y * t + (1/2) * g * t^2
Where:
Δy is the change in height (75.0 meters)
v0y is the vertical component of the initial velocity (which we want to find)
t is the time taken (which we'll solve for)
g is the acceleration due to gravity (approximately 9.8 m/s^2)
Since the balloon is launched straight up, the initial vertical velocity (v0y) is positive. The final velocity at the highest point is zero.
At the highest point, the equation becomes:
0 = v0y - g * t
We can solve this equation for t. Rearranging the equation, we get:
t = v0y / g
Now, since the balloon hits the kite on its upward trajectory, we know that the time taken to reach the maximum height is half of the total time of flight.
Therefore, the total time of flight is:
2t = v0y / g
Now, let's consider the horizontal motion. Since there are no horizontal forces acting on the balloon, the horizontal velocity is constant throughout the motion. Hence, the horizontal component of the launch velocity will remain the same.
Now that we have the total time of flight, we can use it to find the horizontal component of the launch velocity. We know that the horizontal distance traveled is zero because the kite is at the same horizontal position as the balloon.
Therefore, we can write:
0 = v0x * (2t)
Since the horizontal distance traveled is zero, we can solve this equation for v0x (horizontal component of the launch velocity).
Hence, the launch velocity of the balloon (v0) is equal to the square root of [(v0x)^2 + (v0y)^2].
Let's substitute the values we know into the equation and solve for v0:
v0y = 12.2 m/s
g = 9.8 m/s^2
First, let's find the total time of flight:
2t = v0y / g
2t = 12.2 m/s / 9.8 m/s^2
2t = 1.2449 seconds
t = 0.62245 seconds
Now, let's use the total time of flight to find v0x:
0 = v0x * (2 * 0.62245 s)
v0x = 0 m/s
Since v0x = 0, the horizontal component of the launch velocity is zero.
Finally, let's calculate the launch velocity by taking the square root of [(v0x)^2 + (v0y)^2]:
v0 = sqrt((0 m/s)^2 + (12.2 m/s)^2)
v0 = sqrt(148.84 m^2/s^2)
v0 ≈ 12.2 m/s
Therefore, the launch velocity of the balloon was approximately 12.2 m/s.