A 1.4kg lab cart is compressed 10cm against a spring with spring constant k=150N/m.
If the cart is released and the spring transfers all of its energy into the cart, what will be the cart’s speed after the spring is fully extended?
To find the cart's speed after the spring is fully extended, we need to apply the principle of conservation of energy. The energy stored in the compressed spring is transferred to the cart as it extends.
First, let's calculate the potential energy stored in the spring when it is compressed. The formula for potential energy of a spring is given by:
Potential energy (U) = (1/2) * k * x^2
Where:
k is the spring constant (150N/m)
x is the displacement (10cm or 0.1m)
So, substituting the values into the formula, we have:
U = (1/2) * 150 * (0.1)^2
U = 0.75 Joules
According to the conservation of energy, this potential energy will be converted into kinetic energy when the spring is fully extended. The formula for kinetic energy is:
Kinetic energy (K) = (1/2) * m * v^2
Where:
m is the mass of the cart (1.4kg)
v is the velocity (speed) of the cart
Equating the potential energy to the kinetic energy, we have:
U = K
0.75 = (1/2) * 1.4 * v^2
Now, we can solve for the velocity (v):
1.5 = 1.4 * v^2
v^2 = 1.5 / 1.4
v^2 = 1.0714
Taking the square root of both sides, we get:
v ≈ 1.035 m/s
Therefore, the cart's speed after the spring is fully extended will be approximately 1.035 m/s.