Question
A ski jumper acquires a speed of 108.6 km/hr by racing down a steep hill. He then lifts off into the air from a horizontal ramp. Beyond this ramp, the ground slopes downward at an angle of 45 degrees. Assuming the skier is in free-fall motion after he leaves the ramp, at what distance d down the slope does the skier land?
Answers
GPT 3.5
To determine the distance the skier lands, we need to find the horizontal distance traveled by the skier while in free-fall motion.
First, we need to convert the initial speed of the skier from kilometers per hour to meters per second. We can do this by dividing the speed by 3.6:
108.6 km/hr / 3.6 = 30.17 m/s (rounded to two decimal places)
Next, we can break down the initial speed into its horizontal and vertical components. The vertical component of the speed will determine how long the skier will be in the air.
The vertical component of the speed can be found using trigonometry:
Vertical component = speed * sin(angle)
Vertical component = 30.17 m/s * sin(45 degrees)
Vertical component = 30.17 m/s * 0.7071
Vertical component = 21.35 m/s (rounded to two decimal places)
The horizontal component of the speed will determine the distance the skier will travel:
Horizontal component = speed * cos(angle)
Horizontal component = 30.17 m/s * cos(45 degrees)
Horizontal component = 30.17 m/s * 0.7071
Horizontal component = 21.35 m/s (rounded to two decimal places)
Since the skier is in free-fall motion, the time of flight can be determined using the vertical component of speed and the acceleration due to gravity (9.8 m/s^2):
Time of flight = (2 * vertical component) / acceleration due to gravity
Time of flight = (2 * 21.35 m/s) / 9.8 m/s^2
Time of flight = 4.35 seconds (rounded to two decimal places)
We can then calculate the horizontal distance traveled by the skier using the horizontal component of speed and the time of flight:
Distance = horizontal component * time of flight
Distance = 21.35 m/s * 4.35 s
Distance = 92.77 meters (rounded to two decimal places)
Therefore, the skier lands at a horizontal distance of approximately 92.77 meters down the slope.
First, we need to convert the initial speed of the skier from kilometers per hour to meters per second. We can do this by dividing the speed by 3.6:
108.6 km/hr / 3.6 = 30.17 m/s (rounded to two decimal places)
Next, we can break down the initial speed into its horizontal and vertical components. The vertical component of the speed will determine how long the skier will be in the air.
The vertical component of the speed can be found using trigonometry:
Vertical component = speed * sin(angle)
Vertical component = 30.17 m/s * sin(45 degrees)
Vertical component = 30.17 m/s * 0.7071
Vertical component = 21.35 m/s (rounded to two decimal places)
The horizontal component of the speed will determine the distance the skier will travel:
Horizontal component = speed * cos(angle)
Horizontal component = 30.17 m/s * cos(45 degrees)
Horizontal component = 30.17 m/s * 0.7071
Horizontal component = 21.35 m/s (rounded to two decimal places)
Since the skier is in free-fall motion, the time of flight can be determined using the vertical component of speed and the acceleration due to gravity (9.8 m/s^2):
Time of flight = (2 * vertical component) / acceleration due to gravity
Time of flight = (2 * 21.35 m/s) / 9.8 m/s^2
Time of flight = 4.35 seconds (rounded to two decimal places)
We can then calculate the horizontal distance traveled by the skier using the horizontal component of speed and the time of flight:
Distance = horizontal component * time of flight
Distance = 21.35 m/s * 4.35 s
Distance = 92.77 meters (rounded to two decimal places)
Therefore, the skier lands at a horizontal distance of approximately 92.77 meters down the slope.