Two spherical objects, both of mass m and radius R, have center-to-center separa

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. 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. 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.

Explanation / Answer

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

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.