A ski jumper leaves a slope at an angle of 20.0 degrees above the horizontal direction. She lands 3.50
seconds later at a point 20.0 meters below her take-off point.
(a) What was her initial speed?
(b) How far does she travel horizontally?
h(t) = vt - 1/2 g t^2
3.5v - 4.9*3.5^2 = -20
v = 11.436
That v is the vertical speed at takeoff. So, the entire initial speed is
11.436/sin20° = 33.437
Use that speed and
d = vt to get the horizontal distance traveled.
d = 33.437 * 3.5 = 117.028
oops. the horizontal speed is 33.437 cos20°
make that fix...
To find the answers to the given questions, we need to break down the motion into horizontal and vertical components. Let's start with the horizontal component.
The horizontal motion of the ski jumper is not affected by gravity since there is no acceleration in the horizontal direction. So, the horizontal speed remains constant throughout the motion.
(a) To find the initial speed (v₀), we need to use the horizontal component of motion. The horizontal distance traveled can be calculated using the equation:
distance (d) = horizontal speed (v₀) × time (t)
Given:
distance (d) = ? (to be calculated)
horizontal speed (v₀) = ?
time (t) = 3.50 seconds
Since the horizontal speed remains constant, we can use the formula:
v₀ = d / t
Rearranging the formula, we have:
d = v₀ × t
We know that the horizontal distance traveled is not given directly, but we can calculate it using the vertical distance traveled (20.0 meters) and the angle of launch (20.0 degrees).
The horizontal distance (d) can be found using the equation:
d = vertical distance / tan(angle)
d = 20.0 meters / tan(20.0 degrees)
Calculating it, we get:
d = 57.18 meters (rounded to two decimal places)
Now, we can plug in the values we have into the equation:
v₀ = d / t
v₀ = 57.18 meters / 3.50 seconds
Calculating it, we get:
v₀ ≈ 16.34 m/s (rounded to two decimal places)
So, the initial speed of the ski jumper is approximately 16.34 m/s.
(b) To find the horizontal distance traveled, we already have the value from part (a), which is 57.18 meters. So, the ski jumper travels 57.18 meters horizontally.