A novice skier, starting from rest, slides down a frictionless 33.0° incline whose vertical height is 127 m. How fast is she going when she reaches the bottom?
To find the speed of the skier when she reaches the bottom of the incline, we can use the principles of energy conservation.
1. First, we need to calculate the potential energy of the skier at the top of the incline using the formula:
PE = m * g * h
Where:
- PE is the potential energy
- m is the mass of the skier
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
- h is the vertical height of the incline (127 m)
Since the mass of the skier is not given, we can assume a value of 1 kg to simplify the calculation.
PE = 1 kg * 9.8 m/s^2 * 127 m = 1244.6 J
2. Next, we can calculate the speed of the skier at the bottom of the incline using the conservation of energy equation:
PE + KE = KE'
Where:
- KE is the initial kinetic energy (at rest, so KE = 0)
- KE' is the final kinetic energy (when the skier reaches the bottom)
- PE is the potential energy
Since the skier starts from rest, the initial kinetic energy is zero.
0 + PE = KE'
Therefore:
KE' = PE
3. Finally, we can calculate the speed of the skier using the formula:
KE' = (1/2) * m * v^2
Where:
- KE' is the final kinetic energy
- m is the mass of the skier (1 kg, as assumed earlier)
- v is the speed of the skier (what we want to find)
Rearranging the equation:
v^2 = (2 * KE') / m
Substituting the known values:
v^2 = (2 * 1244.6 J) / 1 kg
v^2 = 2489.2 J/kg
Taking the square root to solve for v:
v = √2489.2 J/kg ≈ 49.9 m/s
Therefore, the skier is going approximately 49.9 m/s when she reaches the bottom of the incline.
To determine how fast the skier is going when she reaches the bottom of the incline, we can use the principles of conservation of energy and the concepts of linear motion.
Here's how we can break down the problem and find the answer step-by-step:
1. Begin by defining the given values:
- The angle of the incline, θ = 33.0°
- The vertical height of the incline, h = 127 m
- The initial speed or velocity, u = 0 m/s (since the skier starts from rest)
- The final speed or velocity, v (what we need to find)
2. Determine the gravitational potential energy at the start and the kinetic energy at the end:
- Gravitational potential energy (GPE) at the top, PE = m * g * h, where m is the mass of the skier and g is the acceleration due to gravity (9.8 m/s²).
- Kinetic energy (KE) at the bottom, KE = (1/2) * m * v², where v is the speed of the skier at the bottom.
Since energy is conserved (no energy lost due to friction or other factors), we can equate the GPE at the top with the KE at the bottom:
PE = KE
m * g * h = (1/2) * m * v²
Simplifying, g * h = (1/2) * v²
v² = 2 * g * h
3. Plug in the values and calculate:
v² = 2 * 9.8 m/s² * 127 m
v² = 2490.8 m²/s²
4. Finally, take the square root of both sides to find the velocity:
v = √(2490.8 m²/s²)
v ≈ 49.91 m/s
Therefore, the skier will be going approximately 49.91 m/s when she reaches the bottom of the incline.
The inclined plane forms a rt. triangle:
Y(ver) = 127 m,
A = 33 deg = angle bet. hor side and hyp.
d(hyp) = Y / sinA = 127 / sin33 = 233.2 m.
Vf^2 = Vo^2 + 2gd,
Vf^2 = 0 + 2 * 9.8 * 233.2,
Vf^2 = 4570.7,
Vf = 67.6 m/s = Final velocity.
V=sq.root(2gh)=sq.root-- 2 x 9.8m/s x 127m=49.89m/s
Angle does not really matter.