To find the initial velocity of the ball thrown horizontally from a cliff, we can break the problem down into two components: the horizontal motion and the vertical motion. Since the ball is thrown horizontally, its initial vertical velocity is zero. Let's analyze the situation step by step.
Understanding the Motion Components
The ball is subjected to two types of motion:
- Horizontal Motion: The ball travels horizontally at a constant speed since there is no acceleration in the horizontal direction (ignoring air resistance).
- Vertical Motion: The ball falls under the influence of gravity, which accelerates it downward at approximately 9.81 m/s².
Vertical Motion Analysis
We know the ball takes 5 seconds to hit the ground. We can use the formula for vertical displacement:
Displacement (s) = Initial Velocity (u) × Time (t) + (1/2) × Acceleration (a) × Time²
Since the initial vertical velocity (u) is 0 (the ball is thrown horizontally), the equation simplifies to:
s = (1/2) × g × t²
Substituting the values:
- g = 9.81 m/s² (acceleration due to gravity)
- t = 5 seconds
Calculating the vertical displacement:
s = (1/2) × 9.81 m/s² × (5 s)² = (1/2) × 9.81 × 25 = 122.625 m
Horizontal Motion Analysis
Next, we need to find the horizontal distance traveled by the ball during the 5 seconds. The horizontal distance (d) can be expressed as:
d = Initial Velocity (v₀) × Time (t)
We need to find the initial velocity (v₀). To do this, we can relate the horizontal distance to the angle of projection. The angle of 37 degrees gives us a relationship between the vertical and horizontal components of the motion.
Using the Angle of Projection
The angle of 37 degrees can be used to find the relationship between the vertical and horizontal distances:
tan(θ) = Opposite / Adjacent
Here, the opposite side is the vertical distance (122.625 m), and the adjacent side is the horizontal distance (d). Thus:
tan(37°) = 122.625 m / d
Calculating tan(37°), we find it is approximately 0.7536. Therefore:
0.7536 = 122.625 m / d
Rearranging gives:
d = 122.625 m / 0.7536 ≈ 162.5 m
Finding the Initial Velocity
Now that we have the horizontal distance (d), we can find the initial velocity:
v₀ = d / t
Substituting the values:
v₀ = 162.5 m / 5 s = 32.5 m/s
Final Result
The initial velocity of projection of the ball is approximately 32.5 m/s. This value represents the speed at which the ball was thrown horizontally from the cliff before it began its descent.
