The 1994 Winter Olympics included the aerials competition in skiing. In this event skiers speed down a ramp that slopes sharply upward at the end. The sharp upward slope launches them into the air, where they perform acrobatic maneuvers. In the women's competition, the end of a typical launch ramp is directed 63° above the horizontal. With this launch angle, a skier attains a height of 10.2 m above the end of the ramp. What is the skier's launch speed?
Determine the vertical speed necessary to get to ten meters high. Do that by finding the time to top first.
vericalspeedattop=vi - g time
solve for time in terms of vi, then
h= vi( time) - 1/2 g time^2
solve for vi
To find the skier's launch speed, we need to determine the vertical speed necessary to reach a height of 10.2 meters. We can start by finding the time it takes for the skier to reach the top of the trajectory.
The vertical speed at the top can be calculated using the formula:
vertical speed at top = initial vertical speed (vi) - gravitational acceleration (g) x time
Let's solve this equation to find the time to the top:
vi - g x time = vertical speed at top
Since the skier reaches a height of 10.2 meters, we can use another equation to relate the height to the initial vertical speed and time:
height = vi x time - (1/2)g x time^2
Substituting the known values, we have:
10.2 = vi x time - (1/2)g x time^2
Now, solve this equation for time in terms of vi:
10.2 = vi x time - (1/2)g x time^2
Rearranging the equation:
0.5g x time^2 - vi x time + 10.2 = 0
This equation is a quadratic equation in terms of time. Plug in the known values: g = 9.8 m/s^2 and solve the equation to find the value of time.
Once we find the value of time, we can substitute it back into the first equation to find the initial vertical speed (vi). Rearrange the equation to solve for vi:
vertical speed at top = vi - g x time
Now we have enough information to calculate the skier's launch speed.