A car is released from rest on a frictionless inclined plane. (Figure: twitpic com/e8nkhj) Calculate the momentum pi at the end of the plane in terms of measured quantities x, y, l, and m. Assume θ is very small so that h/l ~= y/x. (Hint: use conservation of energy and K = 1/2mv^2 = p^2/2m
The answer is pi = m(2gyL_i/x)^.5
L_i is supposed to be L subscript i
Could you explain how exactly to derive this equation based on kinetic energy and conservation of energy?
To derive the equation for the momentum at the end of the plane (pi), we can use the principles of kinetic energy and conservation of energy.
1. Start by considering the initial conditions: The car is released from rest, so its initial velocity (Vi) is zero. The initial height (hi) at the top of the plane is y.
2. Next, consider the final conditions: The car reaches the bottom of the inclined plane. The final height (hf) at the bottom of the plane is 0.
3. Apply the conservation of energy principle: The total energy at the top (Ei) is equal to the total energy at the bottom (Ef).
Ei = Ef
Potential Energy at the top (PEi) + Kinetic Energy at the top (KEi) = Potential Energy at the bottom (PEf) + Kinetic Energy at the bottom (KEf)
4. Calculate the potential energy at the top and bottom of the plane:
PEi = m * g * hi (m: mass of the car, g: acceleration due to gravity)
PEf = m * g * hf (hf = 0)
5. Calculate the kinetic energy at the bottom of the plane (KEf):
KEf = (1/2) * m * vf^2 (vf: final velocity at the bottom)
6. The car starts from rest, so the initial kinetic energy (KEi) is zero.
KEi = 0
7. Rewrite the conservation of energy equation using the calculated values:
PEi + KEi = PEf + KEf
m * g * hi + 0 = m * g * hf + (1/2) * m * vf^2
8. Simplify the equation:
g * hi = (1/2) * vf^2
9. Rearrange the equation to solve for vf:
vf = (2 * g * hi)^0.5
10. The final velocity (vf) is the momentum at the end of the plane (pi) divided by the mass of the car (m):
pi / m = vf
pi = m * vf
pi = m * (2 * g * hi)^0.5
11. Substitute hi with the given expression h/l * x from the assumption:
pi = m * (2 * g * (h/l * x))^0.5
12. Simplify the equation further:
pi = m * (2 * g * h * x / l)^0.5
Finally, we arrive at the equation: pi = m * (2 * g * h * x / l)^0.5