Answer :
In order to find out how fast the skateboarder was moving before hitting the ramp, we need to use the principles of energy conservation. We'll be using the standard units of kilograms (kg) for mass, meters (m) for height and distance, meters per second (m/s) for velocity, and Joules (J) for energy. Here are the steps:
**Step 1: Calculate the Potential Energy (gravitational) of the skateboarder when they stop.**
The potential energy (PE) of an object that is raised above the ground can be calculated with the formula:
\[ PE = m \cdot g \cdot h \]
where:
- \( m \) is the mass of the object, in this case, the skateboarder's weight, which is 62 kg.
- \( g \) is the acceleration due to gravity, which is approximately 9.81 m/s².
- \( h \) is the height above ground, which is 2 m in this case.
Substituting in the values we have:
\[ PE = 62 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 2 \, \text{m} \]
\[ PE = 1218.44 \, \text{J} \]
The unit for energy is Joules (J).
**Step 2: Calculate the Kinetic Energy the skateboarder had before they started up the ramp.**
Due to energy conservation, the Kinetic Energy (KE) at the bottom of the ramp (just before going up) is equal to the Potential Energy (PE) at the top of the ramp, since there are no other energy transfers (like work done against friction or air resistance) to account for in this problem.
\[ KE = PE \]
\[ KE = 1218.44 \, \text{J} \]
The unit for energy is Joules (J).
**Step 3: Calculate the velocity using the formula for Kinetic Energy.**
The formula for kinetic energy is:
\[ KE = \frac{1}{2} m v^2 \]
Where \( v \) is the velocity.
We can rearrange this formula to solve for \( v \):
\[ v = \sqrt{\frac{2 \cdot KE}{m}} \]
Substituting in the values we have for KE and \( m \):
\[ v = \sqrt{\frac{2 \cdot 1218.44 \, \text{J}}{62 \, \text{kg}}} \]
\[ v = \sqrt{\frac{2436.88}{62}} \, \text{m/s} \]
\[ v = \sqrt{39.303 \, \text{m/s}^2} \]
\[ v \approx 6.27 \, \text{m/s} \]
The unit for velocity is meters per second (m/s).
When we round this value to the nearest whole number, we get:
\[ v \approx 6 \, \text{m/s} \]
So before hitting the ramp, the skateboarder was moving at approximately 6 meters per second.
