Eduqas Physics for A Level Year 2: Student Bk

96 Eduqas A Level Physics Year 2: Component 2 Example: using gravitational potential The mean radius of Mars is 3.39 × 10 6 m and the value of g at its surface is 3.71 N kg –1 . Neglecting any resistive effects, calculate: (a) the distance that a rocket will rise if it is launched straight up from the surface of Mars with a speed, u , of 3200 m s –1 . (b) the escape velocity of a rocket from the surface of Mars. Answer (a) Since energy is conserved, initial E K + initial E P = E K at highest point + E P at highest point. So 1 2 mu 2 + – GMm r Mars = 0 + – GMm r max in which r max is the distance from the centre of Mars at which the missile runs out of KE. Dividing through by GMm and re-arranging: 1 r max = 1 r Mars – u 2 2 GM therefore 1 r max = 1 r Mars – u 2 2 g surf r 2 Mars Note how we’ve just overcome a dirty trick: not being given the mass, M , of Mars! See Study point. So 1 r max = 1 3.39 × 10 6 m – (3200 m s –1 ) 2 2 × 3.71 m s –1 × (3.39 × 10 6 m) 2 = 1.75 × 10 –7 m –1 Thus r max = 5.72 × 10 6 m and r max  −  r Mars = 2.33 × 10 6 m (b) We use energy conservation as in (a), but with less effort, as this time not only the ‘final’ KE, but also the ‘final’ PE, is zero ( r = ∞). Initial KE + initial PE = KE at infinity + PE at infinity. So 1 2 mu 2 + – GMm r Mars = 0 + 0 So u esc = 2 GM r Mars = 2 g surf × r 2 Mars r Mars = √ 2 g surf r Mars = 5020 m s −1 (e) g and V due to more than one body When the field at a point P arises from more than one body: • The field strength, g , is the vector sum of the field strengths at P that would arise from each body by itself. • The potential, V , is the scalar sum of the potentials that would arise from each body by itself. Example Fig. 2.6.20 shows the alignment of Sun, Moon and Earth during a total eclipse of the Sun. Calculate the resultant field strength and potential at the Moon due to the Earth and Sun. 1.51 × 10 11 m 3.70 × 10 8 m M Sun = 1.99 × 10 30 kg M Earth = 5.97 × 10 24 kg Moon Fig. 2.6.20 Sun, Moon and Earth during a total eclipse Study point The value of g at the surface, g surf is given by g surf = GM r 2 Mars so we can write g surf r 2 Mars in place of GM . x y 7.78 × 10 11 m S P J Fig. 2.6.21 Sun and Jupiter Study point A body placed at P in Self-test 2.6.20 wouldn’t remain at rest (if P is a fixed point relative to the galaxy). For one thing, Jupiter (and the Sun) are in mutual orbit; they don’t stay put. 2.6.20 Self-test Locate the point P between the Sun and Jupiter at which their resultant g is zero. (See Fig 2.6.21.) First determine the ratio x y and hence the values of x and y . (Masses of Sun and Jupiter: 1.99 × 10 30 kg , 1.90 × 10 27 kg. )