To solve for the required horizontal speed, we can use the kinematic equations of projectile motion:
1. d = v_i t + (1/2) a t^2 (for vertical motion)
2. d = v_f t (for horizontal motion)
3. v_f = v_i + a t (for vertical motion)
where:
d = distance
v_i = initial velocity
v_f = final velocity
a = acceleration (in this case, due to gravity and is -9.8 m/s^2)
t = time
We can use equation 1 to solve for the time it takes for the ski jumper to reach the peak of his jump:
25 = v_i t + (1/2) (-9.8) t^2
Simplifying and solving for t, we get:
t = sqrt((50/v_i))
Next, we can use equation 2 to solve for the time it takes for the ski jumper to travel the desired horizontal distance:
135 = v_f t
Substituting v_f from equation 3, we get:
135 = (v_i - 9.8t) t
Simplifying and substituting t from the first equation, we get:
135 = (v_i - 9.8sqrt((50/v_i))) sqrt((50/v_i))
Squaring both sides and simplifying, we get a quadratic equation in terms of v_i:
4.9v_i^2 - 135^2v_i + 4(50^2) = 0
Solving for v_i using the quadratic formula, we get:
v_i = 31.9 m/s
Therefore, the ski jumper must leave the jump with a horizontal speed of 31.9 m/s to travel a distance of 135 meters while starting at a height of 25 meters.
