A model car of mass 8.0 kg slides down a frictionless ramp into a spring with spring constant k = 5.5 kN/m .

(a) The spring experiences a maximum compression of 20 cm. Determine the height of the initial release point.
(b) Calculate the speed of the model car when the spring has been compressed 12 cm.

m g h = 1/2 k x^2 ... h = k x^2 / (2 m g)

h = 5.5E3 N/m * (.20 m)^2 / (2 * 8.0 kg * 9.8 m/s^2) = ? m

forgot part (b)

m g h - (1/2 k x^2) = 1/2 m v^2 ... v^2 = [m g h - (1/2 k x^2)] * 2 / m

v^2 = [(8.0 * 9.8 * h) - (1/2 * 5.5E3 * .12^2)] * 2 / 8.0 = ? (m/s)^2

To solve this problem, we can use the conservation of mechanical energy. The initial potential energy of the car at the top of the ramp is converted to the maximum potential energy stored in the compressed spring.

(a) To determine the height of the initial release point, we can equate the gravitational potential energy of the car at the top of the ramp to the potential energy stored in the compressed spring. The equation is:

mgh = (1/2)kx^2

where:
m = mass of the car (8.0 kg)
g = acceleration due to gravity (9.8 m/s^2)
h = height of the initial release point (to be determined)
k = spring constant (5.5 kN/m converted to N/m = 5.5 × 10^3 N/m)
x = maximum compression of the spring (20 cm = 0.2 m)

Substituting the values into the equation:

8.0 × 9.8 × h = (1/2) × 5.5 × 10^3 × (0.2)^2

Simplifying and solving for h:

78.4h = 220

h ≈ 2.80 m

Therefore, the height of the initial release point is approximately 2.80 meters.

(b) To calculate the speed of the car when the spring has been compressed 12 cm, we can use the conservation of mechanical energy again. The initial gravitational potential energy of the car is converted into both the kinetic energy of the car and the potential energy stored in the compressed spring.

Using the same equation as before, but with x = 0.12 m:

mgh = (1/2)kx^2 + (1/2)mv^2

where:
v = velocity of the car (to be determined)

Substituting the known values:

8.0 × 9.8 × 2.80 = (1/2) × 5.5 × 10^3 × (0.12)^2 + (1/2) × 8.0 × v^2

Solving for v^2:

219.52 = 3960 + 4v^2

4v^2 = 219.52 - 3960

v^2 = -3772.48

Since velocity cannot be negative, we will ignore the negative value. Taking the square root:

v ≈ 17.93 m/s

Therefore, the speed of the model car when the spring has been compressed 12 cm is approximately 17.93 meters per second.

To solve this problem, we can use the principle of conservation of mechanical energy. We'll consider the potential energy and kinetic energy at different points in the system.

(a) To determine the height of the initial release point, we'll start by finding the potential energy at the maximum compression point of the spring and equating it to the initial potential energy when the car is released from a certain height.

Step 1: Find the potential energy at the maximum compression point of the spring:
At maximum compression, the spring is compressed by 20 cm or 0.2 m. We can calculate the potential energy stored in the spring at this point using the formula:

Potential Energy (PE) = 0.5 * k * x^2

where k is the spring constant and x is the compression of the spring.

PE = 0.5 * (5.5 kN/m) * (0.2 m)^2
= 0.5 * 5.5 * 10^3 * 0.04
= 110 J

Step 2: Equate the potential energy at the maximum compression point to the initial gravitational potential energy:
The initial potential energy of the car is equal to its mass times the gravitational acceleration (g) times the height from which it is released.

Let's denote the height of the initial release point as h.

Potential Energy (PE) = m * g * h

where m is the mass of the car and g is the acceleration due to gravity (approximately 9.8 m/s^2).

PE = (8.0 kg) * (9.8 m/s^2) * h
= 78.4 h J

Setting the potential energy at maximum compression equal to the initial potential energy:

110 J = 78.4 h J

Solving for h:

h = 110 J / 78.4 J
= 1.40 m

Therefore, the height of the initial release point is 1.40 meters.

(b) To calculate the speed of the model car when the spring has been compressed 12 cm or 0.12 m, we'll find the potential energy at this point and convert it into kinetic energy.

Step 1: Find the potential energy at the 12 cm compression point:
Using the same formula as in part (a):

Potential Energy (PE) = 0.5 * k * x^2

PE = 0.5 * (5.5 kN/m) * (0.12 m)^2
= 0.5 * 5.5 * 10^3 * 0.0144
= 39.6 J

Step 2: Convert potential energy to kinetic energy:
The potential energy at the 12 cm compression point will be completely converted to kinetic energy when the car reaches this point.

Potential Energy (PE) = Kinetic Energy (KE)

KE = 39.6 J

The kinetic energy can be calculated using the formula:

Kinetic Energy (KE) = 0.5 * m * v^2

where m is the mass of the car and v is the speed of the car.

0.5 * (8.0 kg) * v^2 = 39.6 J

Solving for v:

v^2 = (39.6 J) / (4 kg)
v^2 = 9.9 m^2/s^2

Taking the square root:

v ≈ √(9.9 m^2/s^2)
v ≈ 3.15 m/s

Therefore, the speed of the model car when the spring has been compressed 12 cm is approximately 3.15 m/s.