A 7000-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 = 3800 N/m. After the first spring compresses a distance of 29.7 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 49.5 cm after first contacting the two-spring system. Find the car's initial speed.
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 total potential energy stored in the compressed springs when the car comes to rest.
First, let's find the potential energy stored in the first spring when it compresses by a distance of 29.7 cm (or 0.297 m). The potential energy stored in a spring is given by the formula:
PE = (1/2) * k * x^2
where PE is the potential energy, k is the spring constant, and x is the compression or extension of the spring.
Using this formula, the potential energy stored in the first spring is:
PE1 = (1/2) * k1 * x1^2
where x1 = 0.297 m and k1 = 1700 N/m.
PE1 = (1/2) * 1700 N/m * (0.297 m)^2
= 71.8345 N * m
Now, let's find the potential energy stored in both springs when the car comes to rest at a total compression of 49.5 cm (or 0.495 m).
The potential energy stored in the second spring is given by:
PE2 = (1/2) * k2 * x2^2
where x2 = 0.495 m and k2 = 3800 N/m.
PE2 = (1/2) * 3800 N/m * (0.495 m)^2
= 1857.975 N * m
Since the two springs act in series, the total potential energy stored in the system is the sum of the potential energies stored in each spring:
Total PE = PE1 + PE2
= 71.8345 N * m + 1857.975 N * m
= 1929.8095 N * m
This total potential energy is equal to the initial kinetic energy of the car. The kinetic energy is given by the formula:
KE = (1/2) * m * v^2
where KE is the kinetic energy, m is the mass of the car, and v is the initial speed of the car.
Since the car comes to rest, the final kinetic energy (KE_final) is zero. Therefore, we have:
KE_initial = KE_final + Total PE
0 = (1/2) * 7000 kg * v^2 + 1929.8095 N * m
Solving this equation for v, we can find the initial speed of the car.