Some kid (mass 20 kg) jumps with a velocity of 3.2 m/s horizontally off some steps (height 0.55 m) onto his heavy duty skateboard (mass 3 kg, height 0.03 m), which is initially at rest. How fast will he be skating? I don't where to start.

To find out how fast the kid will be skating, we can use the principle of conservation of mechanical energy. According to this principle, the total mechanical energy of a system remains constant if no external forces act on it.

Let's break down the problem into steps:

Step 1: Find the potential energy of the kid at the top of the steps.
The potential energy (PE) of an object is given by the equation PE = m * g * h, where m is the mass of the object, g is the acceleration due to gravity (approximately equal to 9.8 m/s^2), and h is the height.
PE = 20 kg * 9.8 m/s^2 * 0.55 m

Step 2: Find the potential energy of the skateboard when the kid jumps on it.
PE = 3 kg * 9.8 m/s^2 * 0.03 m

Step 3: Find the kinetic energy of the system after the kid jumps on the skateboard. The kinetic energy (KE) is given by the equation KE = (1/2) * m * v^2, where m is the mass of the object and v is its velocity.
KE = (20 kg + 3 kg) * v^2 / 2, where v is the velocity of the system after the kid jumps on the skateboard.

Step 4: Set the initial potential energy equal to the final kinetic energy.
PE at the top of the steps + PE of the skateboard = KE after jumping on the skateboard
(20 kg * 9.8 m/s^2 * 0.55 m) + (3 kg * 9.8 m/s^2 * 0.03 m) = (23 kg) * (v^2) / 2

Step 5: Solve for v.
Multiply both sides of the equation by 2 and divide both sides by 23 kg to isolate v^2.
v^2 = [(20 kg * 9.8 m/s^2 * 0.55 m) + (3 kg * 9.8 m/s^2 * 0.03 m)] * 2 / 23 kg
v^2 ≈ 5.146
Taking the square root of both sides, we find v ≈ 2.27 m/s.

So, the kid will be skating at a speed of approximately 2.27 m/s.

To find out how fast the kid will be skating after jumping onto the skateboard, we can use the principle of conservation of energy. This principle states that the total energy before the jump is equal to the total energy after the jump.

Let's break down the problem step by step:

Step 1: Calculate the potential energy before the jump.
The potential energy is given by the formula: PE = m * g * h, where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height.

For the kid:
PE_kid = 20 kg * 9.8 m/s² * 0.55 m
PE_kid = 107.8 J

For the skateboard:
PE_skateboard = 3 kg * 9.8 m/s² * 0.03 m
PE_skateboard = 0.882 J

The total potential energy before the jump (PE_total) is the sum of the potential energy of the kid and skateboard:
PE_total = 107.8 J + 0.882 J
PE_total = 108.682 J

Step 2: Calculate the velocity of the kid before landing on the skateboard.
The kid's initial horizontal velocity is given as 3.2 m/s.

Step 3: Calculate the kinetic energy after the jump.
The kinetic energy is given by the formula: KE = 0.5 * m * v², where m is the mass and v is the velocity.

For the kid:
KE_kid = 0.5 * 20 kg * (3.2 m/s)²
KE_kid = 102.4 J

For the skateboard:
The skateboard is initially at rest, so its kinetic energy is zero: KE_skateboard = 0 J.

The total kinetic energy after the jump (KE_total) is the sum of the kinetic energy of the kid and skateboard:
KE_total = 102.4 J + 0 J
KE_total = 102.4 J

Step 4: Apply the principle of conservation of energy.
According to the principle of conservation of energy, the total energy before the jump (PE_total) is equal to the total energy after the jump (KE_total).

PE_total = KE_total
108.682 J = 102.4 J

Step 5: Solve for the final velocity (v_skateboard).
To find the final velocity of the skateboard, we need to rearrange the equation and solve for v_skateboard.

Final velocity:
v_skateboard = √(2 * (KE_total / m_skateboard))

m_skateboard = mass of the skateboard

Plugging in the values:
v_skateboard = √(2 * (102.4 J / 3 kg))
v_skateboard ≈ √(68.2667 m²/s²)
v_skateboard ≈ 8.26 m/s

Therefore, the kid will be skating at a speed of approximately 8.26 m/s after landing on the skateboard.