To determine the maximum height reached by a stone when thrown at an angle of 45 degrees with the horizontal, we need to consider the projectile motion. In projectile motion, an object moves in a curved path under the influence of gravity.
Assuming there is no air resistance, we can break down the initial velocity of the stone into its horizontal and vertical components. Since the stone is thrown at a 45-degree angle, the initial velocity is equally divided between the horizontal and vertical directions.
The vertical component of the initial velocity (V_y) can be determined using trigonometry: V_y = V_initial * sin(angle)
Similarly, the horizontal component of the initial velocity (V_x) can be determined as: V_x = V_initial * cos(angle)
Since the stone reaches its maximum height when the vertical component of its velocity becomes zero, we can use the vertical motion equations to find the time it takes for this to occur. The vertical motion equation for displacement is given by:
Δy = V_y * t - (1/2) * g * t^2
Where Δy is the change in vertical position (maximum height), V_y is the vertical component of the initial velocity, g is the acceleration due to gravity (approximately 9.8 m/s^2), and t is the time.
At the maximum height, the vertical component of the velocity becomes zero, so we can set V_y to zero in the equation:
0 = V_y - g * t
Solving this equation for t, we get: t = V_y / g
Now, we can substitute the value of V_y: t = (V_initial * sin(angle)) / g
Once we have the time it takes for the stone to reach the maximum height, we can find the maximum height (h) using the equation:
h = V_y * t - (1/2) * g * t^2
Substituting the values, we have: h = (V_initial * sin(angle)) * t - (1/2) * g * t^2
Now, let's assume the initial velocity (V_initial) of the stone. For simplicity, let's assume it to be 10 m/s. Thus, we can substitute the values into the equation:
V_initial = 10 m/s angle = 45 degrees g = 9.8 m/s^2
V_y = V_initial * sin(angle) = 10 m/s * sin(45 degrees) = 10 m/s * (sqrt(2) / 2) = 5 * sqrt(2) m/s
t = (V_initial * sin(angle)) / g = (10 m/s * sin(45 degrees)) / 9.8 m/s^2 ≈ 0.721 s
h = (V_initial * sin(angle)) * t - (1/2) * g * t^2 = (10 m/s * sin(45 degrees)) * 0.721 s - (1/2) * 9.8 m/s^2 * (0.721 s)^2 ≈ 2.55 meters
Therefore, the stone will reach a maximum height of approximately 2.55 meters when thrown at an angle of 45 degrees with the horizontal.