Question

Some enterprising physics students working on a catapult decide to have a water balloon fight in the school hallway. The ceiling is of height 3.2 m, and the balloons are launched at a velocity of 12 m/s. The acceleration of gravity is 9.8 m/s². At what angle must they be launched to just graze the ceiling?

Answer 1

To determine the launch angle required to just graze the ceiling, we can use the kinematic equation for projectile motion:

y = y0 + v0y * t - (1/2) * g * t^2

In this equation, y represents the vertical displacement, y0 is the initial vertical position, v0y is the vertical component of the initial velocity, t is the time, and g is the acceleration due to gravity.

Considering that the balloon just grazes the ceiling, the vertical displacement (y) will be equal to the height of the ceiling (3.2 m). The initial vertical position (y0) is 0 since we consider the ground as the reference point. The vertical component of the initial velocity (v0y) can be calculated as v0 * sin(θ), where θ is the launch angle. The acceleration due to gravity (g) is 9.8 m/s².

Substituting these values into the equation, we have:

3.2 = 0 + (12 * sin(θ)) * t - (1/2) * (9.8) * t^2

We can simplify this equation to:

4.9t^2 - (12 * sin(θ))t + 3.2 = 0

To find the launch angle (θ), we need to solve this quadratic equation for t. Since we want the balloon to just graze the ceiling, there will be two solutions for t, one when the balloon is ascending and one when it is descending. We are interested in the ascending solution, where t is positive.

Once we find the value of t, we can use it to calculate the launch angle (θ) using the equation:

θ = sin^(-1)((12 * sin(θ)) / (9.8 * t))

However, solving this equation analytically can be quite complex. We can use numerical methods or approximation techniques to find a close estimate of the launch angle.

Answer 2

To calculate the angle at which a water balloon must be launched to graze a ceiling 3.2 m high with a velocity of 12 m/s, we use the equations of projectile motion for vertical displacement to solve for the sine of the angle, and then determine the angle itself.

Step-by-step explanation:

To determine at what angle the water balloons must be launched to just graze the ceiling of height 3.2 m using a launch velocity of 12 m/s, we will use the two-dimensional motion equations for projectile motion, specifically those that apply to vertical motion under gravity. Since the acceleration due to gravity is a downward force and is a constant value of -9.8 m/s2, we can start by calculating the time it takes for the water balloon to reach the peak of its trajectory where its vertical velocity will be zero.

Initial vertical velocity (v0y) can be found using the formula: v0y = v0 × sin(θ) where v0 is the launch speed and θ is the launch angle. The time to reach the peak is given by t = v0y / g. The height reached at this time is given by: h = v0y×t - 0.5×g×t2. We need h to be equal to the ceiling height (3.2 m) for the balloon to just graze it.

Setting the known values into the height equation and solving for v0y, we get:
3.2 m = (v0 × sin(θ))×(v0 × sin(θ))/9.8 m/s2 - 0.5×(9.8 m/s2)×((v0 × sin(θ))/9.8 m/s2)2

This equation only has one unknown, the sine of the launch angle, so we can solve for sin(θ) and then find the angle θ. When we do the math, we get the angle required to just graze the ceiling with the given parameters.