A fox fleeing from a hunter encounters a 0.625 m tall fence and attempts to jump it. The fox jumps with an initial velocity of 7.75 m/s at an angle of 45.0°, beginning the jump 1.82 m from the fence. By how much does the fox clear the fence? Treat the fox as a particle.
Y = ver = 7.75sin45 = 5.48m/s.
(Vf)^2 = (Vi)^2 + 2gd = 0,
(5.48)^2 + 2(-9.8)d = 0,
30.03 - 19.6d = 0,
-19.6d = -30.03,
d(up) = -30.03 / -19.6 = 1.53m.
Margin = 1.53 - 0.625 = 0.91m.
To determine how much the fox clears the fence, we need to calculate the horizontal distance it travels before it reaches the fence.
First, we need to find the time it takes for the fox to reach the fence. We can use the horizontal component of the velocity for this calculation. The horizontal component of the velocity (Vx) can be calculated using the formula:
Vx = V * cosθ
where V is the initial velocity (7.75 m/s) and θ is the angle (45.0°). Plugging in the values, we get:
Vx = 7.75 m/s * cos(45.0°)
Vx = 7.75 m/s * 0.7071
Vx ≈ 5.48 m/s
Now, we can calculate the time it takes for the fox to reach the fence using the equation:
distance = velocity * time
The distance is the horizontal distance the fox needs to cover, which is 1.82 m. Plugging in the values, we have:
1.82 m = 5.48 m/s * time
Solving for time:
time = 1.82 m / 5.48 m/s
time ≈ 0.332 s
Now that we have found the time it takes for the fox to reach the fence, we can calculate the vertical distance it travels during this time. We can use the equation:
distance = initial velocity * time + (1/2) * acceleration * time^2
Since the fox is treated as a particle and there is no mention of external forces affecting it, we can consider the vertical acceleration to be that of gravity, which is approximately 9.8 m/s^2.
The initial vertical velocity (Vy) can be calculated using the formula:
Vy = V * sinθ
Plugging in the values:
Vy = 7.75 m/s * sin(45.0°)
Vy = 7.75 m/s * 0.7071
Vy ≈ 5.48 m/s
Now, we can calculate the vertical distance the fox travels:
distance = 5.48 m/s * 0.332 s + (1/2) * (-9.8 m/s^2) * (0.332 s)^2
Simplifying the equation:
distance ≈ 0.918 m
Therefore, the fox clears the fence by approximately 0.918 meters.