A water balloon is launched at a building 24m away with an initial velocity of 18 m/s at an angle of 50° above the horizontal. At what height will the balloon strike the building?
First, we can find the time it takes for the projectile (the water balloon) to reach the building by using the horizontal component of the motion.
The horizontal component of the velocity can be found using the formula:
v_x = v * cos(θ)
v_x = 18 m/s * cos(50°) ≈ 11.58 m/s
Now, we can find the time it takes to reach the building by dividing the distance by the horizontal component of the velocity:
t = (distance) / (horizontal velocity)
t = 24 m / 11.58 m/s ≈ 2.07 s
Now that we know the time it takes to reach the building, we can find the height at which the balloon strikes the building using the vertical component of the motion.
The vertical component of the velocity can be found using the formula:
v_y_initial = v * sin(θ)
v_y_initial = 18 m/s * sin(50°) ≈ 13.79 m/s
Now we can calculate the height using the equation of motion for the vertical displacement (assuming that the initial height is 0):
y = v_y_initial * t - 0.5 * g * t^2
where g = 9.81 m/s^2 (acceleration due to gravity)
y = 13.79 m/s * 2.07s - 0.5 * 9.81 m/s^2 * (2.07s)^2
y ≈ 28.53 m - 0.5 * 9.81 m/s^2 * 4.28 s^2
y ≈ 28.53 m - 21.01 m
y ≈ 7.52 m
So, the height at which the balloon strikes the building is approximately 7.52 meters.
To find the height at which the water balloon will strike the building, we need to break down the motion into horizontal and vertical components.
Step 1: Find the horizontal component of the initial velocity:
The horizontal component of the initial velocity can be found using the equation:
Vx = V * cos(angle)
Given:
Initial velocity (V) = 18 m/s
Angle (θ) = 50°
Using the formula to find the horizontal component:
Vx = 18 m/s * cos(50°)
Vx ≈ 11.57 m/s (rounded to two decimal places)
Step 2: Find the time it takes for the water balloon to reach the building:
The time taken can be found using the horizontal distance and horizontal component of velocity:
Time (t) = Distance / Horizontal velocity
Given:
Distance (d) = 24 m
Horizontal velocity (Vx) ≈ 11.57 m/s
Calculating the time:
t = 24 m / 11.57 m/s
t ≈ 2.07 seconds (rounded to two decimal places)
Step 3: Find the height at which the balloon will strike the building:
The vertical motion can be calculated using the equation for vertical displacement:
Vertical displacement (d) = Vertical initial velocity * time - (1/2) * g * t^2
Given:
Vertical initial velocity (Vy) = V * sin(angle)
Acceleration due to gravity (g) = 9.8 m/s^2 (assuming negligible air resistance)
Time (t) ≈ 2.07 seconds (from step 2)
Calculating the vertical displacement:
Vy = 18 m/s * sin(50°)
Vertical displacement (d) = Vy * t - (1/2) * g * t^2
Substituting the values:
d = (18 m/s * sin(50°)) * 2.07 s - (1/2) * 9.8 m/s^2 * (2.07 s)^2
d ≈ 15.93 m (rounded to two decimal places)
Therefore, the water balloon will strike the building at a height of approximately 15.93 meters.
To solve this problem, we can break down the initial velocity of the water balloon into horizontal and vertical components.
The horizontal component of the velocity can be found using the formula:
Vx = V * cos(θ)
where V is the initial velocity (18 m/s) and θ is the angle above the horizontal (50°).
Vx = 18 m/s * cos(50°) = 18 m/s * 0.6428 = 11.57 m/s
The vertical component of the velocity can be found using the formula:
Vy = V * sin(θ)
Vy = 18 m/s * sin(50°) = 18 m/s * 0.766 = 13.8 m/s
Now we can use the kinematic equations to find the height at which the balloon will strike the building.
We know that the horizontal distance (x) is 24m and the horizontal velocity (Vx) is 11.57 m/s.
Using the formula:
x = Vx * t
where t is the time of flight.
Rearranging the formula to solve for time, we get:
t = x / Vx = 24 m / 11.57 m/s = 2.073 seconds
Now, using the vertical motion equation:
y = Vy * t - (1/2) * g * t^2
where y is the height at which the balloon will strike the building, Vy is the vertical velocity (13.8 m/s), t is the time of flight (2.073 seconds), and g is the acceleration due to gravity (approx. 9.8 m/s^2).
Plugging in the values, we get:
y = (13.8 m/s) * (2.073 s) - (1/2) * (9.8 m/s^2) * (2.073 s)^2
y = 28.6 m - 20.3 m = 8.3 m
Therefore, the water balloon will strike the building at a height of 8.3 meters.