DYNAMICS:
a skier starts at the top of a 30m hill ta an anle of 20 degrees. the skier goes straight down the firctionless slope. How long will it take the skier to get to the bottom?
To find the time it takes for the skier to get to the bottom of the hill, we can use the principles of dynamics and kinematics. Let's break down the problem into components:
1. Weight of the skier:
The weight of an object can be calculated using the formula: weight = mass × acceleration due to gravity. However, since we're only interested in the acceleration component of gravity along the slope, we need to find the effective weight acting down the slope. The effective weight is given by: weight along the slope = weight × sin(θ), where θ is the angle of the slope.
2. Acceleration of the skier:
The net force acting on the skier along the slope is the component of weight along the slope. Using Newton's second law of motion, we have: net force = mass × acceleration along the slope. Thus, acceleration along the slope can be found as: acceleration = net force / mass.
3. Displacement of the skier:
Since the slope is frictionless, there is no horizontal force acting on the skier. Therefore, the displacement is only along the vertical direction, which is equal to the height of the hill (30m).
4. Time taken:
To determine the time taken, we can use the kinematic equation: displacement = initial velocity × time + 0.5 × acceleration × time^2. Considering that the skier starts from rest, the initial velocity is zero. Rearranging the equation, we get: time = √(2 × displacement / acceleration).
Putting it all together, here's how we can find the time it takes for the skier to reach the bottom:
1. Calculate the weight along the slope:
weight along the slope = weight × sin(θ)
2. Determine the acceleration along the slope:
acceleration = net force / mass
3. Substitute the values into the kinematic equation:
time = √(2 × displacement / acceleration)
Let's perform the calculations using the given information:
- Mass, m = (Assume a specific value)
- Acceleration due to gravity, g = 9.8 m/s^2
- Angle of the slope, θ = 20 degrees
- Height of the hill, displacement = 30m
By plugging in the values and solving the equation, you can find the time it takes for the skier to get to the bottom of the hill.