A skateboarder starts up a 1.30m high, 25.0° ramp at a speed of 6.70m/s. The skateboard wheels roll without friction. How far from the end of the ramp does the skateboarder touch down?

To determine how far from the end of the ramp the skateboarder touches down, we can use the principles of projectile motion. Here's how you can solve the problem step by step:

Step 1: Break down the motion into horizontal and vertical components. In this case, the initial velocity (6.70 m/s) can be divided into two components: the horizontal component (v_x) and the vertical component (v_y).

The horizontal component (v_x) remains constant throughout the motion since there is no acceleration in the horizontal direction.
So, v_x = v * cos(theta), where v is the initial velocity (6.70 m/s) and theta is the angle of the ramp (25.0°).

The vertical component (v_y) changes due to the influence of gravity. It can be calculated using the equation v_y = v * sin(theta), where v is the initial velocity (6.70 m/s) and theta is the angle of the ramp (25.0°).

Step 2: Find the time of flight (t), which is the time taken by the skateboarder to reach the ground. We can use the vertical component to find the time of flight. The formula for time is given by t = 2 * v_y / g, where g is the acceleration due to gravity (9.8 m/s²).

Step 3: Calculate the horizontal distance traveled (d) by the skateboarder. This can be found using the equation d = v_x * t.

Now, let's plug in the given values and calculate the answer:

v_x = 6.70 m/s * cos(25.0°)
v_y = 6.70 m/s * sin(25.0°)
t = 2 * v_y / g
d = v_x * t