a 55 kg skateboarder on a 3 kg board starts coasting on level ground at 8 m/sec. Let k=3.2 kg/sec. About how far will the skater coast before reaching a complete stop?
To determine the distance the skater will coast before coming to a complete stop, we need to use the principle of kinetic energy.
The initial kinetic energy of the system (skater and board) is given by the formula:
KE_initial = (1/2) * (m_skater + m_board) * v_initial^2
where
m_skater is the mass of the skateboarder (55 kg),
m_board is the mass of the board (3 kg),
and v_initial is the initial velocity (8 m/s).
Substituting the given values, we have:
KE_initial = (1/2) * (55 + 3) * (8)^2
Next, we need to find the final kinetic energy when the skater comes to a complete stop. At this point, the final kinetic energy will be zero.
KE_final = 0
Now, let's introduce a new variable, x, as the distance the skater will coast. We know that there is a relationship between the initial and final kinetic energies and the distance traveled.
According to the work-energy principle, the work done on an object is equal to the change in its kinetic energy:
Work_done = KE_initial - KE_final
Since we know the work done is equal to the force of friction multiplied by the distance traveled (W = F * x), we can rewrite the equation:
F * x = KE_initial - KE_final
The force of friction is given by the equation:
Frictional force = k * (m_skater + m_board)
where k is the constant (3.2 kg/sec).
Substituting the values, we have:
(3.2 * (55 + 3)) * x = (1/2) * (55 + 3) * (8)^2
Now we can solve for x:
(3.2 * 58) * x = (1/2) * 58 * 64
cancel out the common factor of 58:
3.2 * x = (1/2) * 64
Simplifying further:
3.2 * x = 32
Dividing both sides by 3.2:
x = 10
Therefore, the skateboarder will coast for approximately 10 meters before coming to a complete stop.