*Question*

A planet has radius *R*. The escape speed from the surface of the planet is *v*. At what distance from the surface of the planet is the orbital speed 0.5*v*?

A 0.5*R*

B R

C 2*R*

D4*R*

**Answer/Explanation**

Ans: B

ESCAPE SPEED (V_{e})

ORBITAL VELOCITY (V_{0} )

Now given that

\(v_o=0.5v_e\)

hence

\(\sqrt{\frac{GM}{r}} = 0.5 \sqrt{\frac{2GM}{R}} =\frac{1}{2} \sqrt{\frac{2GM}{R}}\)

or

\(\frac{GM}{r}=\frac{1}{4}\times \frac{2GM}{R}\)

\(r =2R\)

hence from surface of planet distance \(= 2R-R =R\)

*Question*

A particle with charge – 2.5 × C moves from point X to point Y due to a uniform electrostatic field. The diagram shows some equipotential lines of the field.

What is correct about the motion of the particle from X to Y and the magnitude of the work done by the field on the particle?

Ans: | Motion of the particle from X to Y | Magnitude of the work done by the field on the particle |

A | uniform linear | 0 J |

B | uniform linear | 1 J |

C | uniformly accelerated | 0 J |

D | uniformly accelerated | 1 J |

**Answer/Explanation**

Ans: D

Force \(F_q = qE\) is from point X towards Y as q is negative. Since E is uniform , hence F is uniform and hence particle accelerate uniformly.

Now

Work done \(W= q \Delta V \)

\(W= q\Delta V = -2.5 \times 10 ^{-6}(700-300)\times 10^3 =-1000\times 10^{-6}=-1 J \)

Magnitude of work done by field \(= |- W|= 1J \)

*Question*

A satellite orbiting a planet moves from orbit X to orbit Y.

What is the change in the kinetic energy and the change in the gravitational potential energy as a result?

**Answer/Explanation**

### Markscheme

C

\(K.E = \frac{1}{2}mv_o^2 where v_o^2 = \frac{GM}{r}\)

\(\therefore K.E =\frac{GMm}{2r}\) hence as \(r \;\uparrow K.E \; \downarrow\)

Potential Energy is given by \(\frac{-GMm}{r}\)

Hence as \(r \; \uparrow \; P.E \;\uparrow \)(since it become less negative)

*Question*

An electron of mass *m*_{e }orbits an alpha particle of mass *m*_{α }in a circular orbit of radius *r*. Which expression gives the speed of the electron?

A. \(\sqrt {\frac{{2k{e^2}}}{{{m_e}r}}} \)

B. \(\sqrt {\frac{{2k{e^2}}}{{{m_a}r}}} \)

C. \(\sqrt {\frac{{4k{e^2}}}{{{m_e}r}}} \)

D. \(\sqrt {\frac{{4k{e^2}}}{{{m_a}r}}} \)

**Answer/Explanation**

## Markscheme

A

Centripetal force = Electrostatic attraction force (since Electron is in circular orbit)

\(\frac{m_ev^2}{r}=\frac{k (e)(2e)}{r^2}\)

Alpha particle is \({He}^{++}\)

Hence

\(v^2 =\frac{k (e)(2e)}{m_er}\)

or

\(v=\sqrt{\frac{k (e)(2e)}{m_er}}=\sqrt{\frac{2k e^2}{m_er}} \)

*Question*

A spacecraft moves towards the Earth under the influence of the gravitational field of the Earth.

The three quantities that depend on the distance *r* of the spacecraft from the centre of the Earth are the

I. gravitational potential energy of the spacecraft

II gravitational field strength acting on the spacecraft

III. gravitational force acting on the spacecraft.

Which of the quantities are proportional to \(\frac{1}{{{r^2}}}\)?

A. I and II only

B. I and III only

C. II and III only

D. I, II and III

**Answer/Explanation**

## Markscheme

C

gravitational potential energy

\(U_g =\frac{-GM_em_s}{r}\)

gravitational field strength

\(E_g =\frac{GM_e}{r^2}\)

gravitational force

\(F_g=\frac{GM_em_s}{r^2}\)

Here We can See gravitational field strength and gravitational force depends on \(r\) as \(\frac{1}{{{r^2}}}\)

*Question*

Four uniform planets have masses and radii as shown. Which planet has the smallest escape speed?

**Answer/Explanation**

### Markscheme

C

Escape velocity is given by

\(v_e=\sqrt{\frac{2GM_p}{R_p}}\)

For smallest \(v_e , M_p \) should be small and \(R_p \) should be big

\(\frac{M_p}{R_p}\)

for \(A = 1\) , For \(B =2\) , for \(C = \frac{1}{2} \) and for \(D =1\)

Hence C will have least escape velocity

*Question*

The graph shows the variation of the gravitational potential *V* with distance *r* from the centre of a uniform spherical planet. The radius of the planet is *R*. The shaded area is *S*.

What is the work done by the gravitational force as a point mass *m* is moved from the surface of the planet to a distance 6*R* from the centre?

A. *m* (*V*2 – *V*1 )

B. *m* (*V*1 – *V*2 )

C. *mS*

D. *S*

**Answer/Explanation**

### Markscheme

B

Gravitational potential is given by

W= -m\Delta V as gravitational field is conservative field and hence work done is independent of path. negative sign because force and dispalcement are in opposite direction

W=- m(V_2-V_1)=m(V_1 -V_2)