To find the speed at which the mass needs to be thrown, we can use the equations of projectile motion. Let's break this down into steps:
Step 1: Convert the angle to radians:
Since the angle is given in degrees, we need to convert it to radians. In radians, 1 degree is equal to π/180 radians. Therefore, the angle of 60 degrees can be converted to radians as follows:
θ = (60 degrees) x (π/180 radians/degree)
θ = π/3 radians
Step 2: Determine the vertical and horizontal components of the velocity:
The initial velocity can be separated into its vertical (Vy) and horizontal (Vx) components. Since the initial speed is the same for both components, we can calculate them using the following equations:
Vy = V_initial × sin(θ)
Vx = V_initial × cos(θ), where V_initial is the initial velocity of the mass.
Step 3: Find the time of flight:
The time of flight is the time the mass takes to reach the wall. We can calculate it using the vertical motion equation:
10 meters = (Vy × t) - (0.5 × g × t^2)
Where g is the acceleration due to gravity (approximately 9.8 m/s^2) and t is the time of flight.
Step 4: Find the horizontal distance traveled during the time of flight:
The horizontal distance (X) traveled by the mass during the time of flight can be calculated using the horizontal motion equation:
X = Vx × t
Step 5: Set up the system of equations and solve for V_initial:
We know that the horizontal distance (X) is 12 meters. By substituting the values from the previous steps into the equation, we can solve for V_initial.
To summarize, the steps to solve this problem are:
1. Convert the angle from degrees to radians.
2. Calculate the vertical and horizontal components of the initial velocity.
3. Use the vertical motion equation to find the time of flight.
4. Use the horizontal motion equation to find the horizontal distance traveled during the time of flight.
5. Set up the system of equations and solve for the initial velocity (V_initial).