A 1000 kg car rolling on a horizontal surface has speed v = 55 km/h when it strikes a horizontal coiled spring and is brought to rest in a distance of 2.2 m. What is the spring stiffness constant of the spring?
To find the spring stiffness constant of the spring, we can use the formula for potential energy stored in a spring:
Potential energy (PE) = 0.5 * k * x^2
Where:
PE is the potential energy stored in the spring
k is the spring stiffness constant
x is the displacement of the spring
In this problem, the car comes to rest in a distance of 2.2 m. This means that the displacement of the spring (x) is 2.2 m. The car's initial kinetic energy is converted into potential energy stored in the spring.
The car's initial kinetic energy (KE) can be calculated using the formula:
KE = 0.5 * m * v^2
Where:
m is the mass of the car
v is the velocity of the car
In this problem, the mass of the car (m) is 1000 kg and the velocity (v) is 55 km/h. We need to convert the velocity from km/h to m/s:
velocity (v) = 55 km/h * (1000 m/1 km) / (3600 s/1 h) = 15.28 m/s
Now we can calculate the initial kinetic energy (KE) of the car:
KE = 0.5 * 1000 kg * (15.28 m/s)^2 = 117104 J
Since the potential energy stored in the spring is equal to the initial kinetic energy of the car, we can set up the equation:
PE = KE = 0.5 * k * x^2
117104 J = 0.5 * k * (2.2 m)^2
Simplifying the equation:
k = (2 * 117104 J) / (2.2 m)^2
k = 106405.45 N/m
Therefore, the spring stiffness constant of the spring is approximately 106405.45 N/m.
To find the spring stiffness constant of the spring, we need to use the concept of conservation of mechanical energy. The mechanical energy of the car-spring system is conserved throughout the process.
First, let's convert the car's speed from km/h to m/s:
v = 55 km/h = (55 * 1000) m/3600 s
v = 15.28 m/s
Now, let's define the variables:
m = mass of the car = 1000 kg
v₀ = initial velocity of the car = 15.28 m/s
d = distance by which the car is brought to rest = 2.2 m
k = spring stiffness constant (what we need to find)
The initial mechanical energy of the car is given by the kinetic energy:
Eᵢ = 0.5 * m * v₀²
The final mechanical energy of the car-spring system is zero, as the car is brought to rest. The final mechanical energy is given by the potential energy of the compressed spring:
E_f = 0.5 * k * d²
According to the principle of conservation of mechanical energy, the initial energy is equal to the final energy:
Eᵢ = E_f
Substituting the values:
0.5 * m * v₀² = 0.5 * k * d²
Now, we can solve for the spring stiffness constant:
k = (0.5 * m * v₀²) / d²
Substituting the given values:
k = (0.5 * 1000 kg * (15.28 m/s)²) / (2.2 m)²
Calculating this expression will give us the spring stiffness constant.