A Nordic jumper goes off a ski jump at an angle of 10.0° below the horizontal, traveling 109.0 m horizontally and 44.0 m vertically before landing. (a) Ignoring friction and aerodynamic effects, calculate the speed needed by the skier on leaving the ramp.
To calculate the speed needed by the skier on leaving the ramp, we can use the principles of projectile motion.
We are given the angle of the jump (10.0° below the horizontal), the horizontal distance traveled (109.0 m), and the vertical distance traveled (44.0 m).
First, let's break down the initial velocity of the skier into horizontal and vertical components:
Horizontal component: The horizontal component of the initial velocity remains constant throughout the motion. Therefore, the initial horizontal velocity (Vx) remains the same as the horizontal distance traveled (109.0 m).
Vertical component: The initial vertical velocity (Vy) can be determined using the vertical distance traveled (44.0 m) and the angle of the jump.
To find Vy, we use the formula:
Vy = V * sin(θ),
where V is the magnitude of the initial velocity and θ is the angle of the jump in radians.
The given angle is in degrees, so we need to convert it to radians by multiplying it by π/180:
θ = 10.0° * (π/180).
Now we can calculate Vy:
Vy = V * sin(10.0° * π/180).
Next, we can use the equations of motion to find the time it takes for the skier to reach the ground. The equations for projectile motion are:
Vertical distance traveled: y = Vy * t + (1/2) * g * t^2,
where y is the vertical distance traveled, Vy is the initial vertical velocity, t is the time, and g is the acceleration due to gravity.
We know the vertical distance traveled is 44.0 m, and g is approximately 9.8 m/s^2. By substituting these values into the equation, we get:
44.0 = Vy * t + (1/2) * (9.8) * t^2.
Now the only unknown in this equation is t, which represents the time taken to reach the ground. We can solve this quadratic equation for t.
Once we have the time, we can use it to find the initial velocity (V) using the equation:
Horizontal distance traveled: x = Vx * t,
where x is the horizontal distance traveled and Vx is the initial horizontal velocity.
We know the horizontal distance traveled is 109.0 m, and we already determined that Vx is also 109.0 m/s.
Finally, we've found the initial velocity required for the skier, which is the magnitude of the initial velocity:
V = √(Vx^2 + Vy^2).
Substituting the known values for Vx and Vy, we can calculate V.
Let's go through the calculations step-by-step to find the final answer.