To determine the initial speed of the stone, we can analyze its motion along the vertical direction. We'll assume there is no air resistance.
Given: Angle above the horizontal (θ) = 53 degrees Maximum height (h_max) = 24 m Acceleration due to gravity (g) ≈ 9.8 m/s² (taking approximate value)
When a projectile reaches its maximum height, its vertical velocity component becomes zero. Using this information, we can find the initial vertical velocity component (v_y) of the stone.
Using the equation of motion for vertical motion:
v_y² = v₀_y² - 2gΔy
Where: v_y is the final vertical velocity component (0 m/s at maximum height) v₀_y is the initial vertical velocity component (what we need to find) g is the acceleration due to gravity (9.8 m/s²) Δy is the change in height (maximum height, 24 m)
0² = v₀_y² - 2 * 9.8 m/s² * 24 m
Rearranging the equation:
v₀_y² = 2 * 9.8 m/s² * 24 m
v₀_y² = 470.4 m²/s²
Taking the square root of both sides:
v₀_y ≈ 21.69 m/s
The initial vertical velocity component is approximately 21.69 m/s.
To find the initial speed of the stone (v₀), we can use the relation between the vertical and horizontal components of the initial velocity:
v₀ = √(v₀_x² + v₀_y²)
Since the stone is thrown upward, the initial horizontal velocity component (v₀_x) remains constant throughout the motion. Therefore, the initial speed of the stone is equal to the magnitude of the initial velocity:
v₀ = √(v₀_x² + v₀_y²) ≈ √(v₀_y²)
v₀ ≈ √(21.69 m/s) ≈ 4.66 m/s
Therefore, the initial speed of the stone is approximately 4.66 meters per second.