On a roller coaster ride the cart is 250 kg. Initially, the cart is traveling at 5 m/s at the upper level, and then 25 once it reaches the flat lower level that is 8 m off the ground. How high up is the upper level?

initial pe+initial ke=final pe+final ke

mgh+1/2 m5^2=mg8+1/2 m 25^2

solve for h

To find the height of the upper level, we can use the principle of conservation of energy. The total mechanical energy of the roller coaster cart is conserved, which means the sum of its potential energy and kinetic energy remains constant throughout the ride.

The total mechanical energy (E) of the cart can be calculated using the formulas:
E = Potential energy + Kinetic energy

The potential energy (PE) of an object is given by the formula:
PE = mgh

where m is the mass (250 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height.

The kinetic energy (KE) of an object is given by the formula:
KE = (1/2)mv^2

where m is the mass (250 kg), and v is the velocity.

At the upper level, the cart is traveling at 5 m/s, so its kinetic energy is:
KE1 = (1/2)(250 kg)(5 m/s)^2

At the lower level, the cart is traveling at 25 m/s, so its kinetic energy is:
KE2 = (1/2)(250 kg)(25 m/s)^2

Since the total mechanical energy is conserved, we have:
E = KE1 + PE1 = KE2 + PE2

At the upper level, the potential energy is PE1 = mgh1, where h1 is the height of the upper level.

At the lower level, the potential energy is PE2 = mgh2, where h2 is the height of the lower level.

Since the lower level is 8 m off the ground, h2 = 8 m.

Therefore, our equation becomes:
KE1 + mgh1 = KE2 + mgh2

Plugging in the values:
(1/2)(250 kg)(5 m/s)^2 + (250 kg)(9.8 m/s^2)h1 = (1/2)(250 kg)(25 m/s)^2 + (250 kg)(9.8 m/s^2)(8 m)

Simplifying the equation will give us the height of the upper level (h1).

To find the height of the upper level, we can use the conservation of mechanical energy.

The initial potential energy (PE_i) at the upper level is equal to the final kinetic energy (KE_f) at the lower level.

The potential energy at a height (h) is given by the equation:

PE = m * g * h

Where:
m = mass of the cart = 250 kg
g = acceleration due to gravity = 9.8 m/s^2

Let's calculate the potential energy at the upper level:

PE_i = m * g * h_i

The initial kinetic energy (KE_i) is given by the equation:

KE = (0.5) * m * v^2

Where:
v = velocity at the upper level = 5 m/s

Let's calculate the initial kinetic energy:

KE_i = (0.5) * m * v^2

At the lower level:
The final kinetic energy (KE_f) is given by the equation:

KE_f = (0.5) * m * v_f^2

Where:
v_f = velocity at the lower level = 25 m/s

Since the potential energy at the upper level is equal to the final kinetic energy at the lower level, we have:

PE_i = KE_f

m * g * h_i = (0.5) * m * v_f^2

We can cancel out the mass (m) from both sides:

g * h_i = (0.5) * v_f^2

Rearranging the equation gives:

h_i = (0.5) * v_f^2 / g

Plugging in the given values:

h_i = (0.5) * (25)^2 / 9.8

Now, let's calculate the value of h_i:

h_i = (0.5) * 625 / 9.8

h_i = 31.25 / 9.8

h_i ≈ 3.19 meters

Therefore, the upper level is approximately 3.19 meters high.