A roller coaster is initially at a height of 40 m above the ground and has an initial velocity of 15 m/s. Using conservation of energy, find the velocity of the roller coaster at a height of 5 m above the ground.
h₁=40 m, h₂=5m, v₁=15 m/s
PE1+KE1 =PE2+KE2
mgh₁+mv ₁²= mgh₂+ mv₂²/2
Solve for v₂
To find the velocity of the roller coaster at a height of 5 m above the ground using conservation of energy, we can use the principle that the total mechanical energy of the system remains constant.
The total mechanical energy of the roller coaster is the sum of its potential energy and kinetic energy:
E = PE + KE
At the initial height of 40 m, the potential energy (PE) of the roller coaster is given by:
PE_initial = m * g * h_initial
where m is the mass of the roller coaster, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h_initial is the initial height.
At the initial velocity of 15 m/s, the kinetic energy (KE) of the roller coaster is given by:
KE_initial = 1/2 * m * v_initial^2
where v_initial is the initial velocity.
Since the total mechanical energy is conserved, at any height h, the total mechanical energy is the sum of the potential energy and kinetic energy at that height:
E = PE + KE = m * g * h + 1/2 * m * v^2
where v is the velocity at height h.
Using conservation of energy, we can equate the initial and final total mechanical energies:
E_initial = E_final
m * g * h_initial + 1/2 * m * v_initial^2 = m * g * h_final + 1/2 * m * v_final^2
Substituting the given values:
m * 9.8 m/s^2 * 40 m + 1/2 * m * (15 m/s)^2 = m * 9.8 m/s^2 * 5 m + 1/2 * m * v_final^2
Simplifying the equation:
392 m * g + 1/2 * 225 m^2/s^2 = 49 m * g + 1/2 * v_final^2
Solving for v_final:
v_final^2 = 392 m * g + 1/2 * 225 m^2/s^2 - 49 m * g
Taking the square root of both sides:
v_final = sqrt(343 m * g + 112.5 m^2/s^2)
Substituting the value of g (approximately 9.8 m/s^2):
v_final = sqrt(343 * 9.8 + 112.5)
Calculating the value:
v_final ≈ sqrt(3361.4 + 112.5) ≈ sqrt(3473.9) ≈ 58.96 m/s
Therefore, the velocity of the roller coaster at a height of 5 m above the ground is approximately 58.96 m/s.