A skier is gliding along at 4.7m/s on horizontal, frictionless snow. He suddenly starts down a 10∘ incline. His speed at the bottom is 15m/s .
What is the length of the incline?
How long does it take him to reach the bottom?
To solve this problem, we can use the principles of kinematics. First, let's find the length of the incline.
Given:
Initial speed of the skier on horizontal snow, u = 4.7 m/s
Final speed of the skier at the bottom, v = 15 m/s
Angle of the incline, θ = 10 degrees
To find the length of the incline, we will use the conservation of energy. The potential energy at the top is equal to the sum of the kinetic energy at the bottom and the work done against gravity.
The potential energy at the top is given by mgh, where m is the mass of the skier, g is the acceleration due to gravity, and h is the height of the incline. The kinetic energy at the bottom is given by (1/2)mv^2.
Since the initial speed on horizontal snow is 4.7 m/s, we can assume that the initial kinetic energy is equal to the kinetic energy at the bottom.
Equating the potential energy at the top with the kinetic energy at the bottom:
mgh = (1/2)mv^2
Canceling the mass (m) from both sides:
gh = (1/2)v^2
Solving for h:
h = (1/2)v^2/g
Using trigonometry, we can express the height of the incline in terms of the length of the incline (L) and the angle (θ):
h = L*sin(θ)
By substituting L*sin(θ) for h in the equation above, we get:
L*sin(θ) = (1/2)v^2/g
Now we can solve for L:
L = (1/2)v^2/(g*sin(θ))
The length of the incline is given by L = (1/2) * (15 m/s)^2 / (9.8 m/s^2 * sin(10 degrees)).
Calculating L:
L = (1/2) * 225 m^2/s^2 / (9.8 m/s^2 * 0.1736)
L ≈ 7.74 meters
Therefore, the length of the incline is approximately 7.74 meters.
To find the time taken to reach the bottom, we can use the equation of motion:
v = u + at
Where:
u = 4.7 m/s (initial velocity)
v = 15 m/s (final velocity)
a = acceleration (due to the incline)
t = time taken
The acceleration a can be calculated using the angle of the incline:
a = g * sin(θ)
Substituting this value into the equation of motion:
v = u + g * sin(θ) * t
Rearranging the equation to solve for time:
t = (v - u) / (g * sin(θ))
Given:
u = 4.7 m/s
v = 15 m/s
θ = 10 degrees
g = 9.8 m/s^2
Substituting these values into the equation:
t = (15 m/s - 4.7 m/s) / (9.8 m/s^2 * sin(10 degrees))
Calculating t:
t ≈ 1.60 seconds
Therefore, it takes approximately 1.60 seconds for the skier to reach the bottom.
To find the length of the incline, we can use the equations of motion. In this case, we can use the equation:
v^2 = u^2 + 2as
where:
v = final velocity (15 m/s)
u = initial velocity (4.7 m/s)
a = acceleration (which is due to gravity and can be calculated using g x sin(θ), where g is the acceleration due to gravity and θ is the angle of the incline)
s = length of the incline (what we are trying to find)
First, let's calculate the acceleration due to gravity:
g = 9.8 m/s^2
Next, we calculate the acceleration along the incline:
a = g x sin(θ)
= 9.8 m/s^2 x sin(10∘)
Now, we can substitute the values into the equation:
15^2 = 4.7^2 + 2 x (9.8 m/s^2 x sin(10∘)) x s
Simplifying the equation:
225 = 22.09 + (19.6 x 0.17364 x s)
225 - 22.09 = 3.40464 x s
202.91 = 3.40464 x s
Finally, we can solve for s:
s = 202.91 / 3.40464
s ≈ 59.63 m
Therefore, the length of the incline is approximately 59.63 meters.
To find the time it takes for the skier to reach the bottom, we can use the equation:
v = u + at
where:
v = final velocity (15 m/s)
u = initial velocity (4.7 m/s)
a = acceleration (9.8 m/s^2 x sin(10∘))
t = time (what we are trying to find)
Substituting the values into the equation:
15 = 4.7 + (9.8 m/s^2 x sin(10∘)) x t
Simplifying the equation:
15 - 4.7 = 9.8 x 0.17364 x t
10.3 = 1.700712 x t
Finally, we can solve for t:
t = 10.3 / 1.700712
t ≈ 6.05 seconds
Therefore, it takes approximately 6.05 seconds for the skier to reach the bottom.
L = (V^2-Vo^2)/2g
a. L = (15^2-4.7^2)/19.6 = 10.4 m.
b. h = L*sin10
h = 10.4*sin10 = 1.8 m.
h = Vo*t + 0.5g*t^2 = 1.8
4.7t + 4.9t^2 = 1.8
4.9t^2 + 4.7t - 1.8 = 0
t = 0.293 s. Use Quadratic Formula.