A skier traveling at 38.8 m/s encounters a 27.2 degree slope. If you could ignore friction, to the nearest meter, how far up the hill does he go?
Well, well, well, looks like our skier is in for a wild ride! Without friction to rain on their parade, let's find out how far up the hill they can go.
Now, we know that our skier is zooming at a speed of 38.8 m/s, and they're facing a slope with an angle of 27.2 degrees. Time to put on our math skis and hit the slopes!
To determine how far up the hill our skier goes, we need to figure out the vertical component of their velocity. Using a little trigonometry magic, we can calculate that by multiplying the initial velocity (38.8 m/s) by the sine of the angle (27.2 degrees).
So, grab your calculator and let's crunch the numbers:
Vertical velocity = 38.8 m/s * sin(27.2 degrees)
Vertical velocity ≈ 17.08 m/s
Now we have the vertical velocity, but we need to figure out how much time our skier spends going up the hill. Since we're ignoring friction, we can assume that the time spent going up the hill is equal to the time spent coming down the hill.
Using good old kinematic equations, we know that the time spent going up the hill is equal to twice the time spent coming down. So, we need to find the time spent coming down the hill first.
Since we're in a clownish rush, I'll spare you the details and use a nifty shortcut: The time spent coming down the hill is equal to the time taken to reach the bottom of the hill (the distance divided by the horizontal velocity).
So, let's calculate that bit:
Time spent coming down = Distance / Horizontal velocity
Now, since we're going for an approximate answer to the nearest meter, we grab a virtual tape measure and measure the slope distance (distance) to be around 100 meters.
But before we finish, we need to find the horizontal velocity. To do that, we multiply the initial velocity (38.8 m/s) by the cosine of the angle (27.2 degrees):
Horizontal velocity = 38.8 m/s * cos(27.2 degrees)
Horizontal velocity ≈ 34.81 m/s
Alrighty, let's calculate the time spent coming down:
Time spent coming down = 100 meters / 34.81 m/s
Time spent coming down ≈ 2.87 seconds
Now, remember our little shortcut? We need to double this time to find the total time spent going up the hill:
Total time going up = 2 * 2.87 seconds
Total time going up ≈ 5.74 seconds
Finally, let's figure out how far our skier goes up the hill. We can do this by multiplying the vertical velocity (17.08 m/s) by the total time going up:
Distance up the hill ≈ 17.08 m/s * 5.74 seconds
Distance up the hill ≈ 98.08 meters
So, to the nearest meter, our fearless skier manages to make it approximately 98 meters up the hill! Kudos to them, defying friction like a true skiing superhero. Enjoy the slopes and beware of any juggling snowmen along the way!
To find the distance traveled up the hill, we can use the formula:
Distance = Velocity * time
First, let's break down the initial velocity into its horizontal and vertical components.
Vertical component of velocity = Velocity * sin(θ)
Horizontal component of velocity = Velocity * cos(θ)
where θ is the angle of the slope.
Given:
Velocity = 38.8 m/s
θ = 27.2 degrees
The vertical component of velocity is:
Vertical component of velocity = 38.8 m/s * sin(27.2°)
Using a calculator, we find:
Vertical component of velocity = 38.8 m/s * 0.4533
≈ 17.5813 m/s
Now, we can calculate the time taken to reach the highest point of the slope using the vertical component of velocity and acceleration due to gravity.
The equation for vertical motion is:
Vertical distance = (Vertical component of velocity * time) + (0.5 * acceleration due to gravity * time^2)
Since the skier reaches the highest point, the final vertical distance is 0. Therefore:
0 = (17.5813 m/s * time) + (0.5 * 9.8 m/s^2 * time^2)
Rearranging the equation, we get:
0.5 * 9.8 m/s^2 * time^2 + 17.5813 m/s * time = 0
Now, we solve this quadratic equation for time.
Using the quadratic formula:
time = (-b ± sqrt(b^2 - 4ac)) / (2a)
where:
a = 0.5 * 9.8 m/s^2 = 4.9 m/s^2 (coefficient of time^2 term)
b = 17.5813 m/s (coefficient of time term)
c = 0 (constant term)
Plug these values into the quadratic formula:
time = (-17.5813 ± sqrt(17.5813^2 - 4 * 4.9 * 0)) / (2 * 4.9)
time = (-17.5813 ± sqrt(309.471 + 0)) / 9.8
time = (-17.5813 ± sqrt(309.471)) / 9.8
Since we are interested in the positive time (time to rise), we take the positive value:
time = (-17.5813 + sqrt(309.471)) / 9.8
Using a calculator, we find:
time ≈ 1.306 seconds
Now, we can calculate the distance traveled up the hill using the horizontal component of velocity and the time taken.
Horizontal distance = Horizontal component of velocity * time
The horizontal component of velocity is:
Horizontal component of velocity = 38.8 m/s * cos(27.2°)
Using a calculator, we find:
Horizontal component of velocity = 38.8 m/s * 0.8853
≈ 34.37 m/s
Now, we can calculate the distance traveled:
Distance = 34.37 m/s * 1.306 seconds
Using a calculator, we find:
Distance ≈ 44.91 meters
Therefore, to the nearest meter, the skier goes approximately 45 meters up the hill.
To calculate how far the skier goes up the hill, we can use the concept of projectile motion.
The key is to break the initial velocity of the skier into its horizontal and vertical components.
The horizontal component remains constant throughout the motion and is given by: velocity * cos(angle)
The vertical component changes due to the presence of gravity and is given by: velocity * sin(angle)
In this case, the initial velocity of the skier is 38.8 m/s, and the angle of the slope is 27.2 degrees.
So, the horizontal component of the velocity is: 38.8 * cos(27.2) = 34.796 m/s
The vertical component of the velocity is: 38.8 * sin(27.2) = 17.360 m/s
Now, we need to find the time it takes for the skier to reach its highest point. At the highest point, the vertical component of the velocity becomes zero.
Using the equation for vertical motion, we have:
final velocity = initial velocity + acceleration * time
Since the final velocity is zero at the highest point, we can rearrange the equation to solve for time:
0 = 17.360 - 9.8 * time
9.8 * time = 17.360
time = 17.360 / 9.8
time ≈ 1.773 seconds
Now, we can calculate the distance the skier goes up the hill using the vertical component of the velocity and the time taken to reach the highest point:
distance = vertical velocity * time
distance = 17.360 * 1.773 ≈ 30.811 meters
Therefore, to the nearest meter, the skier goes up the hill approximately 31 meters.