Ski Ramp Physics: Find Initial Velocity & Injuries

[MarkGhent]

Homework Statement

A Skier leaves the ramp at 25 degrees inclined from the horizontal. The ramp is 4 meters above the ground. The skier lands 100 meters away. The ground itself is sloped from the ramp relating to a 3 4 5 triangle. I need to find the initial velocity off the ramp and the extent of the skiers injuires

Homework Equations

Range=(V^2/g)*sin2(theta)
Time of flight=(2V/g)sin(theta)
x=x(o) +V(o)T

The Attempt at a Solution

I used the range equation to find the initial velocity which i found to be 35.78m/s and then used that in the TOF equation to find the TOF to be 3.08s and then split the velocity into its horizontal and vertical components to get V(x)=32.42m/s V(y)=15.15m/s

I then used V(x) in the x=x(o)+V(o)t to get the x distance to make sure my numbers were right and i got X= 99.8m which is basically 100m. Am I going about this right or right and did the fact that a 3 4 5 triangle was drawn under the 100m distance matter at all?

 

[kamerling]

That equation for the range and time of flight is only valid for ballistic motion that returns to the same altitude. Since you start at 4m altitude and the ground is sloping away they aren't valid here.

you just have to guess a value of v for the initial velocity here. You can than get the time of flight from the distance and v_x. The time of flight and v_y will than give you the altitude at the end of the flight. If the end of the ramp has altitude 0, the ground will have an altitude of -4 - 3/4 x where x is the horizontal distance away from the ramp. (if I uncerstand your remark about a 3 4 5 triangle correctly.)

 

[Moetasim]

Your approach to finding the initial velocity and time of flight using the range equation and time of flight equation is correct. However, the 3 4 5 triangle referenced in the problem does not have any significance in this calculation. It is important to note that these equations assume ideal conditions and do not take into account factors such as air resistance, friction, or the angle of the slope on the ground. These factors can affect the actual distance traveled and the velocity at which the skier lands, potentially leading to injuries.

To accurately assess the extent of the skier's injuries, a more comprehensive analysis would need to be done, taking into account factors such as the skier's mass, the angle at which they land, and the impact force upon landing. This would require further data and calculations, as well as possibly conducting experiments or simulations. It is also important to note that injuries can vary greatly depending on the individual and their physical condition, so it would be difficult to determine a specific extent of injury without more information.

Overall, while the initial velocity can be calculated using the given information, a more thorough analysis would be needed to accurately determine the extent of the skier's injuries.