A 900 kg car rolling on a horizontal surface has speed v = 70 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 (k) of the coiled spring, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position.

Hooke's Law equation can be written as:

F = -kx

Where:
F is the force exerted by the spring
k is the spring stiffness constant
x is the displacement from the equilibrium position

In this case, the car was brought to rest by the spring, so the force exerted by the spring is equal to the force that stopped the car.

The force that stopped the car can be calculated using the principle of momentum conservation:

Initial momentum = Final momentum

The momentum of the car before it was brought to rest can be calculated as:

Initial momentum = mass * initial velocity

Final momentum is zero since the car comes to rest.

Therefore:

mass * initial velocity = 0

Now, let's solve for the force exerted by the spring using the formula:

Force = mass * acceleration

Since the car comes to rest, the acceleration of the car can be calculated using the equation:

v^2 = u^2 + 2as

Where:
u is the initial velocity
v is the final velocity (zero in this case)
a is the acceleration
s is the displacement

Rearranging the equation:

a = (v^2 - u^2) / (2s)

Substituting the given values:

a = (0 - (70 km/h)^2) / (2 * 2.2 m)

Now, we can calculate the force exerted by the spring:

Force = mass * acceleration

Substituting the mass and acceleration values:

Force = 900 kg * [(0 - (70 km/h)^2) / (2 * 2.2 m)]

Finally, we can equate the force exerted by the spring to the force in Hooke's Law equation and solve for the spring stiffness constant:

-kx = 900 kg * [(0 - (70 km/h)^2) / (2 * 2.2 m)]

Rearranging the equation to solve for k:

k = (900 kg * [(0 - (70 km/h)^2) / (2 * 2.2 m)]) / x

Substituting the given displacement value (x = 2.2 m) into the equation will yield the spring stiffness constant (k).