A 3.0 kg object is moving along the x-axis in a region where its potential energy as a function of x is given as U(x) = 4.0x2 , where U is in joules and x is in meters. When the object passes the point x = -0.50 m, its velocity is +2.0 m/s. All forces acting on the object are conservative. Calculate the total mechanical energy of the object Calculate the x-coordinate of any points at which the object has zero kinetic energy. Calculate the magnitude of the momentum of the object at x = 0.60 m. Calculate the magnitude of the acceleration of the object as it passes x = 0.60 m. On the axes below, sketch graphs of the object’s position x versus time t and kinetic energy K versus time t. Assume that x = 0 at time t = 0 . The two graphs should cover the same time interval and use the same scale on the horizontal axes.

Following are the responses to the given points:

Given:

object mass

Potential energy  

Solution:

For point (a)  

Total Energy

For point (b)  

If an object's potential energy is 7 J, it has 0 kinetic energy.

For point (c)  

For point (d)

the velocity is found by  

Calculating the acceleration:

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a) , b) c)

Explanation:

a) The total energy of the object is:

b) The total energy of the object is:

c) The speed of the object is clear in the total energy expression:

The magnitude of the momentum is:

d) Before calculating the acceleration experimented by the object, it is required to determine the net force exerted on it. There is a relationship between potential energy and net force:

Acceleration experimented by the object is:

e) The position of the object versus time is found by solving the following differential equation:

The velocity is obtained by deriving the previous expression:

Speed of the object at is:

The equation of the motion are:

The expression for the kinetic energy of the object is:

Graphics are included below as attachments. (Position versus time, kinetic energy vs time).

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