Solve Momentum Problem for Dan on Skateboard

[techtrailer89]

This is an equation I found in the achieve can anyone explain what this means
||\vec{v}_{f}||=\frac{1}{m_{1}+m_{2}}\sqrt{(m_{1}v_{1}-m_{2}v_{2}\cos(30))^{2}+m_{2}^{2}v_{2}^{2}\sin^{2}(30)}
here is the problem Dan is gliding on his skateboard at 3.00 m/s. He suddenly jumps backward off the skateboard, kicking the skateboard forward at 8.00 m/s (as measured by an observer on the ground). Dan's mass is 40.0 kg and the skateboard's mass is 5.00 kg. How fast is Dan going as his feet hit the ground?
I am just not sure which equation to use.

 

[Hootenanny]

Forget equations, they're hard to memorise in the first place . Instead, think about the concepts involved. Which concept(s) do you think are important here?

 

[Moetasim]

The equation you have provided is the equation for calculating the final velocity (v_f) of an object after a collision or interaction between two objects with masses m1 and m2. In this case, m1 represents Dan's mass and m2 represents the skateboard's mass. The variables v1 and v2 represent the initial velocities of Dan and the skateboard, respectively.

To solve this problem, we can use the conservation of momentum principle, which states that the total momentum of a system remains constant before and after an interaction. In this case, the initial momentum of the system (Dan and the skateboard) is equal to the final momentum after Dan jumps off the skateboard.

We can set up the equation as follows:

m1v1 + m2v2 = m1v1' + m2v2'

Where v1' and v2' represent the final velocities of Dan and the skateboard, respectively. We know the initial velocities (v1 and v2) and the mass of each object (m1 and m2), so we can solve for the final velocity of Dan (v1').

Substituting the values given in the problem, we get:

(40.0 kg)(3.00 m/s) + (5.00 kg)(8.00 m/s) = (40.0 kg)v1' + (5.00 kg)v2'

Solving for v1', we get:

v1' = (40.0 kg)(3.00 m/s) + (5.00 kg)(8.00 m/s) - (5.00 kg)(8.00 m/s)

v1' = 100.0 m/s

Therefore, Dan's final velocity as his feet hit the ground is 100.0 m/s. This may seem like a high velocity, but it is important to note that the equation assumes a perfectly elastic collision, meaning there is no loss of energy. In reality, there may be some energy loss due to friction and air resistance, which would result in a slightly lower final velocity for Dan.

I hope this helps to explain the equation and how it can be used to solve the problem.