A 930 kg car strikes a huge spring at a speed of 22 m/s, compressing the spring 12.0 m.
(a) What is the spring stiffness constant of the spring?
(b) How long is the car in contact with the spring before it bounces off in the opposite direction?
Why has no one answered this?
Because there is no need to
To answer these questions, we need to apply the principles of conservation of energy and the relationship between force, displacement, and stiffness constant.
(a) To determine the spring stiffness constant, 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. The formula is given by:
F = -kx
Where F is the force, k is the spring stiffness constant, and x is the displacement. Rearranging the formula, we get:
k = -F/x
To find the force, we can use conservation of energy. The initial kinetic energy of the car is equal to the potential energy stored in the compressed spring. The formula to calculate kinetic energy is:
KE = 0.5 * m * v^2
Where m is the mass of the car (930 kg) and v is the initial velocity (22 m/s). The formula to calculate potential energy is:
PE = 0.5 * k * x^2
Where k is the spring stiffness constant and x is the compression distance (12.0 m).
Setting the initial kinetic energy of the car equal to the potential energy stored in the spring, we get:
0.5 * m * v^2 = 0.5 * k * x^2
Rearranging the formula, we can solve for k:
k = (m * v^2) / x^2
Plugging in the values, we get:
k = (930 kg * (22 m/s)^2) / (12.0 m)^2
Using a calculator, we can find the spring stiffness constant k.
(b) To find the time the car is in contact with the spring before bouncing off, we need to determine the period of oscillation. The period, T, is the time taken for one complete cycle of the spring oscillation. It can be calculated using the formula:
T = 2π * √(m/k)
Where m is the mass of the car and k is the spring stiffness constant. Once we have the period, we can divide it by 2 to find the time the car is in contact with the spring before bouncing off.