Question

Figure 6.76 shows the path of a comet orbiting a star.

(a) Rank-order the locations on the path in terms of the magnitude of the comet’s momentum at each location, starting with the location at which the magnitude of the momentum is the largest. (b) Rank-order the locations on the path in terms of the comet’s kinetic energy at each location, starting with the location at which the kinetic energy is the largest. (c) Rank-order the locations on the path in terms of the potential energy of the system at each location, largest first. (d) Rank-order the locations on the path in terms of the sum of the kinetic energy and the potential energy of the system at each location, largest first

Short answer

(a) The order of the locations on the path in terms of the magnitude of the comet’s momentum at each location, starting with the location at which the magnitude of the momentum is the largest is $E>D>C>B>A$.

(b) The order of the kinetic energy from largest to least is $E>D>C>B>A$.

(c) The order of the locations on the path in terms of the potential energy of the system at each location from largest to least is $A>B>C>D>E$.

(d) The order the locations on the path in terms of the sum of the kinetic energy and the potential energy of the system at each location from largest to least is $A=B=C=D=E$.

Step-by-step solution

Identification of given data

The given data can be listed below,

• The comet is orbiting around a star.

Concept/Significance of potential energy

Potential energy is related to unbalanced forces poised to perform work, while kinetic energy is linked with momentum.

Energy, like momentum, is preserved, but it has a different form known as potential energy. The kinetic and potential energies are not preserved independently.

Evaluation of Rank-order the locations on the path in terms of the magnitude of the comet’s momentum at each location.

(a)

The momentum of comet is given by,

$p=mv$

Here,mis the mass of the comet,vis the velocity of the comet.

When comet is near the star its speed will increase as the distance between them decreases.

Following the diagram,Ehave greatest speed because its nearer to the star, the momentum is directly proportional to velocity so the momentum will be greatest at pointEfollowed by,D,C,B, andArespectively.

Thus, the order of the locations on the path in terms of the magnitude of the comet’s momentum at each location, starting with the location at which the magnitude of the momentum is the largest is $E>D>C>B>A$.

Evaluation of the Rank-order of the locations on the path in terms of the comet’s kinetic energy at each location.

(b)

The kinetic energy of the comet is given by,

$K.E=\frac{1}{2}m{v}^2$

Here,mis the mass of the comet,vis the velocity of the comet.

From above equation, the kinetic energy of the comet is directly proportional to the velocity of the comet.

So, the kinetic energy of the point having greatest velocity will be maximum. From (a)

It is found that the velocity and momentum f the pointEis highest.

Thus, the order of the kinetic energy from largest to least is $E>D>C>B>A$.

Evaluation of the order the locations on the path in terms of the potential energy of the system at each location

(c)

The gravitational potential energy of the comet is given by,

$U=\frac{-GMm}{r}$

Here, G is the gravitational constant whose value is $6.673\times {10}^{-11}N{m}^2/k{g}^2$, m is the mas of the comet, M is the mass of the star, r is the distance between comet and star.

The potential energy is inversely proportional torwith negative sign.

$U\alpha -\frac{1}{r}$

From above equation, it is seen that the potential energy will increase with increase in distance between star and comet. So, distance between comet and star is greatest at pointAfollowed byB, C, DandE.

Thus, the order of the locations on the path in terms of the potential energy of the system at each location from largest to least is $A>B>C>D>E$.

Evaluation of the order the locations on the path in terms of the sum of the kinetic energy and the potential energy of the system at each location

(d)

The total energy of the comet is given by,

$E=\frac{1}{2}m{v}^2-\frac{GMm}{r}$

From law of conservation of the energy,

$\frac{1}{2}m{v}^2=\frac{GMm}{r}$

From equation it is determined that the energy is equal and constant at all the points of the system.

Thus, the order the locations on the path in terms of the sum of the kinetic energy and the potential energy of the system at each location from largest to least is $A=B=C=D=E$.