016 10.0 points
One of the great dangers to mountain climbers is an avalanche, in which a large mass of snow and ice breaks loose and goes on an essentially frictionless “ride” down a mountainside on a cushion of compressed air.
The acceleration of gravity is 9.8 m/s2 .
30.5◦
Ifyouwereona30.5◦slopeandan avalanche started 303 m up the slope, how much time would you have to get out of the way?
I tried to use the summation of the forces equals mass times acceleration formula but that got me 4.973875957 as the acceleration and then I plugged that in to d=Vit+1/2at^2 but that got me the wrong answer I don't know what else to do
Yes
To find the time you would have to get out of the way of an avalanche on a 30.5° slope, we need to break down the problem into different steps. Let's begin by calculating the acceleration of the avalanche.
1. Determine the acceleration of the avalanche:
The force responsible for the acceleration is the component of gravity acting parallel to the slope. To find this force, we need to calculate the component of gravity along the slope.
The component of gravity parallel to the slope is given by: F_parallel = m * g * sin(θ)
where m is the mass of the avalanche, g is the acceleration due to gravity (9.8 m/s²), and θ is the slope angle (30.5°).
As we don't have the mass of the avalanche, we can assume it cancels out in this case. Thus, the acceleration of the avalanche is the same as the slope:
a = g * sin(θ)
Substituting the values: a = 9.8 m/s² * sin(30.5°), we find the acceleration to be approximately 4.98 m/s².
2. Calculate the time to get out of the way:
Now that we have the acceleration, we can use a kinematic equation to find the time it takes for the avalanche to reach a specific distance.
The equation we can use is: d = v₁ * t + (1/2) * a * t²
where d is the distance, v₁ is the initial velocity, t is the time, and a is the acceleration.
In this case, the initial velocity is 0 since the avalanche starts from rest. The distance traveled by the avalanche is 303 m (given in the question), and the acceleration is 4.98 m/s² (calculated earlier).
Substituting the values, the equation becomes: 303 m = 0 * t + (1/2) * 4.98 m/s² * t²
Simplifying: 0.5 * 4.98 m/s² * t² = 303 m
Rearranging the equation: t² = (2 * 303 m) / (4.98 m/s²)
Calculating: t² ≈ 121.69 s²
Finally, taking the square root of both sides: t ≈ √(121.69 s²), we find that t ≈ 11.03 seconds.
Therefore, you would have approximately 11.03 seconds to get out of the way of the avalanche on a 30.5° slope.