A snowboarder is trying to gap a 14m
wide road. He leaves the 19m high cliff at
10m/s at 30 degrees.
Questions:
1. Does he make it?
2.When the snowboarder is 16m above
the road, what is his speed?
Vo = 10m/s[30o]
Xo = 10*Cos30 = 8.66 m/s
Yo = 10*sin30 = 5 m/s
1. Y = Yo * g*Tr = 0
Tr = -Yo/g = -5/-9.8 = 0.51 s. = Rise time.
h = ho + -(Yo^2)/2g = 19 + -(5^2)/-19.6 = 20.3 m. Above gnd.
h = 0.5g*t^2 = 20.3
4.9t^2 = 20.3
t^2 = 4.14
Tf = 2.03 s. = Fall time.
Dx = Xo*(Tr+Tf) = 8.66 * (0.51+2.03) =
22 m. Which is greater than the required
hor. distance of 14 m.
2. Y^2 = Yo^2 + 2g*h=0 + 19.8*(20.3-16) = 84.28
Y = 9.2 m/s = Ver. component.
V = sqrt(Xo^2+Y^2) m/s
Xo = 8.66 m/s
Y = 9.2 m/s
Solve for V.
To determine if the snowboarder makes it across the 14m wide road, we can calculate the horizontal distance he covers while in the air. We'll also calculate the maximum height reached and the speed when he is 16m above the road.
1. To calculate the horizontal distance covered, we can use the formula for horizontal displacement: d = v * t * cos(theta), where d is the horizontal distance, v is the velocity, t is the time in the air, and theta is the launch angle.
First, let's calculate the time the snowboarder is in the air. Since we know the initial vertical velocity is 10m/s and the angle is 30 degrees, we can decompose the velocity into vertical and horizontal components.
Vertical velocity (v_y) = v * sin(theta)
v_y = 10 * sin(30)
v_y = 5 m/s
The time in the air (t) can be calculated using the formula t = (2 * v_y) / g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
t = (2 * 5) / 9.8
t ≈ 1.02 seconds
Now, let's calculate the horizontal distance (d) covered by the snowboarder.
d = v * t * cos(theta)
d = 10 * 1.02 * cos(30)
d ≈ 8.85 meters
Since the horizontal distance covered (8.85m) is less than the width of the road (14m), the snowboarder does not make it across.
2. To calculate the speed of the snowboarder when he is 16m above the road, we can use the equation for vertical velocity: v_y = v * sin(theta) - g * t, where v_y is the vertical velocity and t is the time.
First, let's calculate the time it takes for the snowboarder to reach a height of 16m above the road from the initial launch.
t = (16 - 0) / (0.5 * g)
t = 16 / 4.9
t ≈ 3.27 seconds
Now, let's calculate the vertical velocity (v_y) at that time.
v_y = v * sin(theta) - g * t
v_y = 10 * sin(30) - 9.8 * 3.27
v_y ≈ -4.6 m/s
The negative sign indicates that the snowboarder is descending at this point. Therefore, the speed of the snowboarder when he is 16m above the road is approximately 4.6 m/s.