A skateboarder shoots off a ramp with a velocity of 6.7 m/s, directed at an angle of 53° above the horizontal. The end of the ramp is 1.0 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.
When the skateboarder reaches the highest point, how far is this point horizontally from the end of the ramp?
I keep seeing the 4.9 where did that come from?
I believe 4.9 is half of A=9.8 inside of the equation Y=Voy(t)+Ay(t2)/2
Why did the + in the equation switch to a -?
To find the horizontal distance from the end of the ramp to the highest point, we can break down the skateboarder's motion into horizontal and vertical components.
First, let's look at the vertical motion. The skateboarder has an initial vertical velocity component of 6.7 m/s * sin(53°) because the velocity is at an angle of 53° above the horizontal. At the highest point of the motion, the vertical velocity would be zero because the skateboarder momentarily stops moving upward before falling downward. We can use this information to find the time taken for the skateboarder to reach the highest point.
Using the kinematic equation for vertical motion:
v_f = v_i + a * t
where v_f is the final vertical velocity (which is 0 m/s at the highest point), v_i is the initial vertical velocity, a is the acceleration (which is the acceleration due to gravity, approximately -9.8 m/s^2), and t is the time taken.
0 = 6.7 m/s * sin(53°) - 9.8 m/s^2 * t
Solving for t, we get:
t = (6.7 m/s * sin(53°)) / 9.8 m/s^2
Now that we know the time taken to reach the highest point, we can find the horizontal distance traveled using the horizontal velocity component.
The horizontal component of the skateboarder's velocity remains constant throughout the motion because there are no horizontal forces acting on the skateboarder. Thus, the horizontal velocity is simply the initial horizontal velocity, which is 6.7 m/s * cos(53°).
To find the horizontal distance traveled, we can use the formula:
Distance = Velocity * Time
Distance = 6.7 m/s * cos(53°) * t
Now we can substitute the value of t that we calculated earlier:
Distance = 6.7 m/s * cos(53°) * ((6.7 m/s * sin(53°)) / 9.8 m/s^2)
Calculating this expression will give you the horizontal distance from the end of the ramp to the highest point.
a)
Vi= 6.7sin(53)
Vi = 5.35
V = Vi - gt
at top 0 = Vi - g t
so t at top = Vi/g
t = .55
y at top = 1 + Vi t - 4.9 t^2
= 1 + 5.35(0.55) - 4.9(.55)^2
y at top = 2.46
b)
u = 6.7 cos 53
x = u t
x = (6.7 cos 53)(.55)
x = 2.22