Wiley Coyote (Carnivouous hungribilous) is chasing the roadrunner (Speediburd cant- catchmi) yet again. While running down the road, they come to a deep gorge, 22 m straight across and 101 m deep. The roadrun- ner launches itself across the gorge at a launch angle of 20◦ above the horizontal, and lands with 2.1 m to spare.
The acceleration of gravity is 9.81 m/s2 .
What was the roadrunner’s launch speed? Ignore air resistance.
Answer in units of m/s
Wiley Coyote launches himself across the gorge with the same initial speed, but at a lower launch angle. To his horror, he is short of the other lip by 1.2 m and falls into the gorge.
What was Wiley Coyote’s launch angle? Answer in units of ◦
123
To find the roadrunner's launch speed, we can use the equation for horizontal distance traveled during projectile motion:
d = v_i * cos(θ) * t
where:
d = horizontal distance
v_i = initial speed
θ = launch angle
t = time of flight
From the problem, we know that the roadrunner lands with 2.1 m to spare, so the horizontal distance traveled is the gorge width minus the extra distance:
d = 22 m - 2.1 m = 19.9 m
Now, we need to calculate the time of flight. The time of flight can be found using the equation:
t = (2 * v_i * sin(θ)) / g
where g is the acceleration due to gravity.
Given:
θ = 20°
g = 9.81 m/s²
Substituting the known values into the equation, we can solve for t:
t = (2 * v_i * sin(20°)) / 9.81
Now, using the equation for horizontal distance, we can solve for the roadrunner's launch speed, v_i:
19.9 = v_i * cos(20°) * [(2 * v_i * sin(20°)) / 9.81]
Simplifying the equation:
19.9 = v_i² * (2 * sin(20°) * cos(20°)) / 9.81
Now, solve for v_i:
v_i² = (19.9 * 9.81) / (2 * sin(20°) * cos(20°))
v_i = sqrt[(19.9 * 9.81) / (2 * sin(20°) * cos(20°))]
Using a scientific calculator, we can calculate the value of v_i to find the roadrunner's launch speed.
To find Wiley Coyote's launch angle, we can use the same equation for horizontal distance:
d = v_i * cos(θ) * t
Given:
d = 22 m - 1.2 m = 20.8 m (short of the other lip)
v_i = same as the roadrunner's launch speed
We can solve for the launch angle, θ, using the equation:
20.8 = v_i * cos(θ) * t
By rearranging the equation:
cos(θ) = 20.8 / (v_i * t)
Now, we substitute the known values:
cos(θ) = 20.8 / (v_i * [(2 * v_i * sin(θ)) / 9.81])
Simplifying the equation:
cos(θ) = (20.8 * 9.81) / (2 * sin(θ) * v_i)
Solve for θ:
θ = arccos[(20.8 * 9.81) / (2 * sin(θ) * v_i)]
Using a scientific calculator, we can calculate the value of θ to find Wiley Coyote's launch angle.