Question

Two spherical objects, both of mass m and radius R, have center-to-center separation 4R and are initially at rest. No external forces act. The spheres accelerate towards one another until they collide.

(a) What is the speed of each object just before the collision? Find the exact answer using

energy methods.

(b) Using the final speed from part (a), and

assuming the acceleration is roughly constant

and equal to the initial acceleration, calculate

an upper limit on the time it takes for the two

objects to collide. (The actual time will be

shorter, because the acceleration increases as

they get closer.)

(c) Calculate a numerical value of the time from

part b, in the case of two bowling balls having

m = 7 kg and R = 0.1 m.

Answer 1

a) Apply conservation of energy

Initial mechanical energy = final mechanical energy

Ui + Ki = Uf + Kf

-G*m^2/(4*R) + 0 = -G*m^2/(2*R) + 2*(1/2)*m*v^2

-G*m^2/(4*R) + G*m^2/(2*R) = m*v^2

G*m^2/(4*R) = m*v^2

G*m/(4*R) = v^2

v = sqrt(G*m/(4*R))

b) acceleration of each object, a = F/m

= G*m^2/(4*R)^2/m

= G*m/(16*R^2)

let t it the time taken.

use, s = u*t + (1/2)*a*t^2

2*R = 0 + (1/2)*(G*m/(16*R^2))t^2

t^2 = 64*R^3/(G*m)

t = 8*sqrt(R^3/(G*m))

c) t = 8*sqrt(0.1^3/(6.67*10^-11*7))

= 1.17*10^4 s