A skateboarder shoots off a ramp with a velocity of 7.9 m/s, directed at an angle of 56° above the horizontal. The end of the ramp is 1.3 m above the ground. Let the x axis be parallel to the ground, the +y direction be vertically upward, and take as the origin the point on the ground directly below the top of the ramp. (a) How high above the ground is the highest point that the skateboarder reaches? (b) When the skateboarder reaches the highest point, how far is this point horizontally from the end of the ramp?
A)
sin(56)*7.9= Vo
Vo= 6.55
y=Vf^2-Vo^2/2a
y=0^2-6.55^2/2(-9.81)
y=-(6.55^2)/2(-9.81)
y= 2.1867
2.1867+ 1.3= 3.5m
To solve this problem, we can analyze the vertical and horizontal motion of the skateboarder separately.
(a) To find the highest point the skateboarder reaches, we need to determine the maximum height reached in the vertical direction. We can use the following kinematic equation:
v^2 = u^2 + 2as
where:
v = final velocity (0 m/s at the highest point)
u = initial velocity (7.9 m/s)
a = acceleration (due to gravity, -9.8 m/s^2)
s = displacement (unknown)
Rearranging the equation, we have:
s = (v^2 - u^2) / (2a)
Substituting the values, we get:
s = (0^2 - 7.9^2) / (2 * -9.8)
s = (-62.41) / (-19.6)
s ≈ 3.19 m
Therefore, the highest point the skateboarder reaches above the ground is approximately 3.19 meters.
(b) To find the horizontal distance from the end of the ramp to the highest point, we need to determine the time it takes for the skateboarder to reach the highest point.
We can use the vertical motion equation:
v = u + at
where:
v = final velocity in the y-direction (0 m/s at the highest point)
u = initial velocity in the y-direction (component of 7.9 m/s in the upward direction)
a = acceleration in the y-direction (due to gravity, -9.8 m/s^2)
t = time (unknown)
Rearranging the equation, we have:
t = (v - u) / a
Substituting the values, we get:
t = (0 - 7.9 * sin(56°)) / (-9.8)
t ≈ 0.96 s
Next, to find the horizontal distance traveled during this time, we use the equation:
d = v * t
where:
d = horizontal distance (unknown)
v = horizontal velocity (component of 7.9 m/s in the x-direction)
t = time (0.96 s)
Substituting the values, we get:
d = 7.9 * cos(56°) * 0.96
d ≈ 4.75 m
Therefore, when the skateboarder reaches the highest point, they are approximately 4.75 meters horizontally from the end of the ramp.