A 250kg rollercoaster cart loaded with people has an initial velocity of 3m/s. Find the velocity of the cart at part A (10m high), B (35m), and C (44m) Assume that friction is negligible.
If the velocity is only 3 m/s at the bottom, no way it will go up 10 meters. Typo? Did you leave out an initial height?
To find the velocities of the rollercoaster cart at different heights, we can use the principle of conservation of mechanical energy. According to this principle, the total mechanical energy of the system remains constant as long as there is no external work and no energy is lost due to friction.
The mechanical energy of the system consists of the sum of the kinetic energy and the potential energy:
E = KE + PE,
Where,
E is the total mechanical energy,
KE is the kinetic energy, and
PE is the potential energy.
At the initial position, the total mechanical energy is:
E_initial = KE_initial + PE_initial,
Since the initial velocity is given and assuming the rollercoaster cart starts from rest at a height of 0m, the initial potential energy is 0. Therefore:
E_initial = KE_initial,
E_initial = (1/2)mv^2,
Where,
m is the mass of the cart (250kg), and
v is the initial velocity (3m/s).
E_initial = (1/2)(250kg)(3m/s)^2 = 1125 Joules.
At any other height h, the total mechanical energy is:
E = KE + PE,
E = (1/2)mv^2 + mgh,
Where,
g is the acceleration due to gravity (9.8 m/s^2),
h is the height.
To find the velocity v at any height h, we can rearrange the equation to solve for v:
E = (1/2)mv^2 + mgh,
(1/2)mv^2 = E - mgh,
v^2 = 2(E - mgh) / m,
v = sqrt(2(E - mgh) / m).
Now, we can find the velocities at different heights:
At part A (height h = 10m):
v_A = sqrt(2(E_initial - mgh_A) / m),
v_A = sqrt(2(1125 J - (250 kg)(9.8 m/s^2)(10m)) / 250 kg),
At part B (height h = 35m):
v_B = sqrt(2(E_initial - mgh_B) / m),
v_B = sqrt(2(1125 J - (250 kg)(9.8 m/s^2)(35m)) / 250 kg),
At part C (height h = 44m):
v_C = sqrt(2(E_initial - mgh_C) / m),
v_C = sqrt(2(1125 J - (250 kg)(9.8 m/s^2)(44m)) / 250 kg).
Calculating the values will give us the answer.