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).

if you already have the answer to (a), then just work out how many periods there are in 1 minute, and that's the answer to (c).

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