A 58 kg skier has a speed of 9.5 m/s at point A. She then encounters a 1.75 m deep dip in the snow's surface that has a circular cross section with a radius of curvature of 12 m.
and the question?
To find the acceleration of the skier at point A, we can use the law of conservation of energy.
The total mechanical energy of the skier at the initial point A consists of kinetic energy (KE) and gravitational potential energy (PE).
KE = (1/2) * mass * velocity^2 --> KE = (1/2) * 58 kg * (9.5 m/s)^2 = 2611.75 J
PE = mass * g * height --> PE = 58 kg * 9.8 m/s^2 * 0 m (since point A is at ground level) = 0 J
The total mechanical energy at point A is the sum of KE and PE:
Total Energy at A = KE + PE = 2611.75 J + 0 J = 2611.75 J
As the skier encounters the dip in the snow's surface at point B, some of the total mechanical energy is converted to potential energy due to the change in height.
Given that the dip has a radius of curvature of 12 m, we can calculate the change in height using the relation between the radius of curvature and the depth of the dip:
Height = Radius of curvature - Depth = 12 m - 1.75 m = 10.25 m
At Point B, the total mechanical energy consists of the potential energy at the height of the dip, as the kinetic energy is converted into potential energy:
Total Energy at B = PE = mass * g * height = 58 kg * 9.8 m/s^2 * 10.25 m = 5822.9 J
The difference in energy between Point A and Point B is due to the conversion of kinetic energy to potential energy:
Energy Difference = Total Energy at B - Total Energy at A = 5822.9 J - 2611.75 J = 3211.15 J
To find the acceleration of the skier at Point A, we can use the energy difference and the equation for work done:
Work = Force * Distance * cos(angle)
Since the force of gravity is acting vertically downwards, the angle between the direction of the force and the displacement is 0 degrees (cos(0) = 1).
Work = Force * Distance
The work done is equal to the energy difference, so we have:
Energy Difference = Force * Distance
To express the force in terms of acceleration, we can use Newton's second law:
Force = mass * acceleration
So, we have:
Energy Difference = mass * acceleration * Distance
Solving for acceleration, we get:
acceleration = Energy Difference / (mass * Distance)
Substituting the given values:
acceleration = 3211.15 J / (58 kg * 12 m) ≈ 4.64 m/s^2
Therefore, the skier's acceleration at Point A is approximately 4.64 m/s^2.