1. Known values:
- Distance covered in each jump (d) = 8.4 m
- Launch angle (θ) = 23°
- Acceleration due to gravity (g) = 9.8 m/s²
Unknown values:
- Initial velocity or takeoff speed (vâ‚€)
- Maximum height above the ground (h)
2. To find the initial velocity or takeoff speed (vâ‚€), we can use the following formula:
v₀ = d / (cosθ * t)
In this case, the time (t) is the total time spent in the air during each jump.
To find the time (t), we can use the formula:
d = v₀ * cosθ * t
Rearranging this formula gives us:
t = d / (v₀ * cosθ)
Now we can substitute this expression for time back into the initial velocity formula:
v₀ = d / (cosθ * (d / (v₀ * cosθ)))
Simplifying this equation gives us:
v₀ = d * v₀ * cosθ / d, or
1 = v₀ * cosθ / 1
Simplifying further gives us:
v₀ = 1 / cosθ
Substituting the value of the launch angle (θ = 23°) into the equation, we get:
v₀ = 1 / cos(23°)
vâ‚€ = 1 / 0.921
Calculating this gives us:
v₀ ≈ 1.085 m/s
So, the initial velocity or takeoff speed is approximately 1.085 m/s.
3. To find the maximum height above the ground (h), we can use the following formula:
h = (v₀ * sinθ)² / (2 * g)
Substituting the values into the equation gives us:
h = (1.085 * sin(23°))² / (2 * 9.8)
Calculating this gives us:
h ≈ 0.085 m
Therefore, the maximum height above the ground is approximately 0.085 m.