A fox fleeing from a hunter encounters a 0.665 m tall fence and attempts to jump it. The fox jumps with an initial velocity of 7.05 m/s at an angle of 45.0°, beginning the jump 2.17 m from the fence. By how much does the fox clear the fence? Treat the fox as a particle.
vertical:
hf=hi+7.05sin45*t-4.9t^2
horizontal:
2.17=7.05cos45*t
solve for t in the horizontal equation, put that in to the verticalequation, then you have its height. Subtract the height of the fence.
To find by how much the fox clears the fence, we first need to break down the initial velocity into its vertical and horizontal components.
Given:
Initial velocity (V₀) = 7.05 m/s
Launch angle (θ) = 45.0°
Distance from the fence (d) = 2.17 m
Height of the fence (h) = 0.665 m
We can find the vertical component of the initial velocity (V₀y) using the sine function:
V₀y = V₀ * sin(θ)
V₀y = 7.05 m/s * sin(45.0°)
V₀y = 4.99 m/s
Now, let's calculate the time it takes for the fox to reach the fence. Since we are treating the fox as a particle, neglecting air resistance, we can use the vertical motion equation:
h = V₀yt - (1/2) * g * t²
Where:
h = Height of the fence = 0.665 m
V₀y = Vertical component of initial velocity = 4.99 m/s
g = Acceleration due to gravity = 9.8 m/s² (assuming downward direction)
t = Time
Rearranging the equation, we get:
(1/2) * g * t² - V₀yt + h = 0
This is a quadratic equation in terms of t. We can solve it using the quadratic formula:
t = (-b ± √(b² - 4ac)) / 2a
where:
a = 1/2 * g
b = -V₀y
c = h
Plugging in the values, we have:
t = (-(-4.99) ± √((-4.99)² - 4 * (1/2 * 9.8) * 0.665)) / (2 * (1/2 * 9.8))
Simplifying further:
t = (4.99 ± √(24.9001 - 2.596)) / 9.8
t = (4.99 ± √(22.3041)) / 9.8
Now we have two possible values for time, one positive and one negative. Since negative time is not meaningful in this context, we consider the positive value.
t ≈ 0.388 seconds (rounded to 3 decimal places)
Next, we can calculate the horizontal distance the fox travels during this time:
d = V₀x * t
Where V₀x is the horizontal component of the initial velocity, and it can be calculated using the cosine function:
V₀x = V₀ * cos(θ)
V₀x = 7.05 m/s * cos(45.0°)
V₀x = 4.99 m/s
d = 4.99 m/s * 0.388 s
d ≈ 1.936 meters (rounded to 3 decimal places)
Finally, the fox clears the fence by the horizontal distance:
Clearance = d - distance from the fence (2.17 m)
Clearance = 1.936 m - 2.17 m
Clearance ≈ -0.234 meters
Therefore, the fox falls short of clearing the fence by approximately 0.234 meters.