A water balloon is fired at 34.0 m/s from a water cannon, which is aimed at an angle of 18° above the ground. The center of the cannon's target (which has a radius of 1.0 m) is painted on the asphalt 42.0 m away from the water cannon.
will the balloon hit the target? justify your response with calculation.
Recall that the range of a missile is given by
R = (v^2 sin2θ)/g
plug in your numbers and see whether 41.0 <= R <= 43.0
To determine whether the water balloon will hit the target, we need to analyze its horizontal and vertical motion.
First, let's break down the initial velocity of the water balloon into horizontal and vertical components. The horizontal component of the velocity (Vx) remains constant throughout the motion and is given by:
Vx = V * cos(θ)
where V is the initial speed of the water balloon (34.0 m/s) and θ is the launch angle (18°).
Vx = 34.0 m/s * cos(18°)
Vx = 34.0 m/s * 0.9511
Vx ≈ 32.34 m/s
The vertical component of the velocity (Vy), on the other hand, changes due to the effect of gravity. Vy is given by:
Vy = V * sin(θ)
where sin(θ) = opposite/hypotenuse = vertical velocity/initial velocity.
Vy = 34.0 m/s * sin(18°)
Vy = 34.0 m/s * 0.3249
Vy ≈ 11.05 m/s
Now, let's calculate the time it takes for the water balloon to reach the target. We'll consider the vertical component of the motion since the target is on the same horizontal level as the cannon.
To find the time of flight (t), we can use the equation:
Vy = g * t
where g is the acceleration due to gravity (9.8 m/s²).
t = Vy / g
t = 11.05 m/s / 9.8 m/s²
t ≈ 1.13 s
Now, we can determine the horizontal distance (x) the water balloon travels during this time:
x = Vx * t
x = 32.34 m/s * 1.13 s
x ≈ 36.61 m
The water balloon will hit the target if its horizontal distance (36.61 m) is less than or equal to the distance to the target (42.0 m).
Thus, the water balloon will hit the target since 36.61 m ≤ 42.0 m.
We have justified this response by calculating the horizontal distance traveled by the water balloon and comparing it to the distance to the target.