A 650 kg roller coaster car has two points of reference given to an observer. Point A, at a height of 120 m, the cart is moving at 3.40 m/s. Later on in the ride, at point B, the cart is moving at 28.2 m/s. Determine the height at point B.
None of the formulas on my formula sheet are helping me with this one.
I think all I need to do is do
B Height = A Height x A Speed over B Speed
But that seems too simple.
Energy relations:
initial total energy=final total energy
This assume zero friction on the roller coaster, a very silly assumption.
1/2*m*3.4^2+m*9.8*120=1/2 m*28.2^2 + mg*final height
find final height.
Your simple solution is way off target in our real world.
Thank you. By the way, there is question is assuming there is no friction, so this helps a lot.
Thank you so much.
To determine the height at point B, you can make use of the conservation of mechanical energy principle, which states that the total mechanical energy of a system remains constant if no external forces are acting on it.
The formula to calculate the total mechanical energy is:
E = KE + PE
Where:
E = Total mechanical energy
KE = Kinetic energy
PE = Potential energy
At point A, the roller coaster has a known height (120 m) and a known velocity (3.40 m/s). So, let's calculate the total mechanical energy at point A.
KE_A = 1/2 * m * v_A^2
Where:
m = mass of the roller coaster car (650 kg)
v_A = velocity at point A (3.40 m/s)
KE_A = 1/2 * 650 * (3.40)^2
Next, calculate the potential energy at point A:
PE_A = m * g * h_A
Where:
g = acceleration due to gravity (approximately 9.8 m/s^2)
h_A = height at point A (120 m)
PE_A = 650 * 9.8 * 120
Finally, find the total mechanical energy at point A:
E_A = KE_A + PE_A
Now, let's determine the height at point B using the conservation of mechanical energy principle.
At point B, the roller coaster has a known velocity (28.2 m/s). We need to find the potential energy at point B and solve for the height.
KE_B = 1/2 * m * v_B^2
Where:
v_B = velocity at point B (28.2 m/s)
KE_B = 1/2 * 650 * (28.2)^2
Next, calculate the potential energy at point B:
PE_B = m * g * h_B
Where:
h_B = height at point B (to be determined)
PE_B = 650 * 9.8 * h_B
Since the total mechanical energy remains constant, we can equate the total mechanical energy at points A and B:
E_A = E_B
KE_A + PE_A = KE_B + PE_B
Now substitute the values we calculated earlier:
1/2 * 650 * (3.40)^2 + 650 * 9.8 * 120 = 1/2 * 650 * (28.2)^2 + 650 * 9.8 * h_B
Solve for h_B, which represents the height at point B. By rearranging the equation and isolating h_B:
h_B = (1/2 * 650 * (28.2)^2 + 650 * 9.8 * 120 - 1/2 * 650 * (3.40)^2) / (650 * 9.8)
Calculate the expression on the right-hand side of the equation to find the height at point B.