A roller coaster powered by a spring launcher follows a frictionless track as shown below. The roller coaster cart has a mass of 500 kg, and the spring launcher has a spring constant of 1500 N/m.
d. What is the fastest speed the cart reaches anywhere on the track?
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To find the fastest speed the cart reaches anywhere on the track, we need to determine the energy of the system at different points. The total mechanical energy of the system is the sum of the potential energy and the kinetic energy.
At the starting point, the cart is compressed against the spring, and all the energy is potential energy. 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 displacement from the equilibrium position (in this case, the compression distance of the spring launcher).
In this case, the potential energy can be expressed as:
PE = (1/2) * 1500 N/m * x^2
Since the total mechanical energy remains constant, the potential energy at the starting point will be converted into kinetic energy at the highest point on the track, where the potential energy is zero and the kinetic energy is at its maximum.
The kinetic energy of an object is given by the equation:
KE = (1/2) m v^2
Where KE is the kinetic energy, m is the mass of the object, and v is the velocity.
Setting the potential energy equal to the kinetic energy, we can equate the equations:
(1/2) * 1500 N/m * x^2 = (1/2) * 500 kg * v^2
Simplifying the equation, we get:
1500 N/m * x^2 = 500 kg * v^2
Dividing both sides by 500 kg, we have:
3 N/m * x^2 = v^2
Taking the square root of both sides, we find:
v = sqrt(3 N/m * x^2)
Since the question asks for the maximum speed anywhere on the track, we need to consider the maximum compression distance of the spring launcher. This value is not specified in the given information, so we cannot determine the exact maximum speed without further details.
Therefore, the fastest speed the cart reaches anywhere on the track is dependent on the maximum compression distance of the spring launcher.