A ball of mass m is shot at a high initial speed v 0 directly toward a large blo

Question

A ball of mass m is shot at a high

initial

speed v0 directly toward a large block of mass

4m

at rest on a frictionless roller-coaster type of track. The

block

is sent toward a hill of height h.

a) assuming the collision between the ball and block

is

perfectly elastic, what are each of their velocities after

the

collision?

b) What is the minimum value of v0 that will allow

the

block to move over the top of the hill?

c) Repeat parts (a) and (b), but this time instead of a

elastic

collision, assume the ball rebounds with a velocity of

-v0/3.

Explanation / Answer

a) conservation of momentum

m v0 = m v1 + 4m v2

v2 = (v0-v1)/4

conservation of energy

1/2 m v0^2 = 1/2 m v1^2 + 1/2 4m v2^2

v0^2 = v1^2 + 4*((v0-v1)/4)^2

v0^2 = v1^2 + (v0^2 - 2 v0 v1 + v1^2)/4

4 v0^2 = 4 v1^2 + v0^2 - 2 v0 v1 + v1^2

5 v1^2 - 2 v0 v1 - 3 v0^2 = 0

v1= -3/5 v0

so v2 = (v0 + 3/5 v0)/4 = 2/5 v0

b) so we want 1/2 mv^2 = m gh

v2 = sqrt(2 g h)

2/5 v0 = sqrt(2 gh)

v0 = 5/2 sqrt(2gh)

c)

conservation of momentum

m v0 = m*-v0/3 + 4m v

4 v0/3 = 4 m v

v = v0/3

so now

v0/3 = sqrt(2gh)

v0 = 3 sqrt(2gh)

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

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