A 8000-kg freight car rolls along rails with negligible friction. The car is brought to rest by a combination of two coiled springs as illustrated in the figure below. Both springs are described by Hooke's law and have spring constants with k1 = 1700 N/m and k2 = 3700 N/m. After the first spring compresses a distance of 29.4 cm, the second spring acts with the first to increase the force as additional compression occurs as shown in the graph. The car comes to rest 45.0 cm after first contacting the two-spring system. Find the car's initial speed.

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To find the car's initial speed, we can use the principle of conservation of mechanical energy. The initial kinetic energy of the car will be equal to the elastic potential energy stored in the two springs when they are compressed.

Let's break down the problem into steps:

Step 1: Calculate the total compression distance of the springs.
The first spring compresses a distance of 29.4 cm (or 0.294 m). The second spring begins to act after this initial compression and the car comes to rest after additional compression of 45.0 cm (or 0.45 m). Therefore, the total compression distance of both springs is 0.294 m + 0.45 m = 0.744 m.

Step 2: Calculate the elastic potential energy stored in the springs.
The elastic potential energy of a spring is given by the formula: E = (1/2) * k * x^2, where E is the potential energy, k is the spring constant, and x is the compression or extension distance.

For the first spring: E1 = (1/2) * k1 * (0.294)^2
For the second spring: E2 = (1/2) * k2 * (0.45 - 0.294)^2 (Using the additional compression distance)

Step 3: Calculate the initial kinetic energy of the car.
Since the initial kinetic energy is equal to the total elastic potential energy stored in the springs:
Initial kinetic energy = E1 + E2

Step 4: Use the relationship between kinetic energy and velocity.
The kinetic energy can be related to the velocity using the equation: K.E. = (1/2) * m * v^2, where K.E. is the kinetic energy, m is the mass of the object, and v is the velocity.

Step 5: Rearrange the equation to solve for the initial velocity.
v = √(2 * Initial kinetic energy / mass)

Step 6: Plug in the values and calculate the initial velocity.
The mass of the car is given as 8000 kg.

Now you can calculate the initial velocity of the car by plugging in the values into the equation in Step 5 and solve for v.

if the springs are in series then the combined spring constant is 1/k=1/1700 + 1/3700

k= (1700*3700)/5400=1168N/m

1/2 k x^2=1/2 m v^2

you have k, x (.45m), and mass m. Solve for v