Imagine two billiard balls on a pool table. Ball A has a mass of 7 kilograms and ball B has a mass of 2 kilograms. The initial velocity of the ball A is 6 meters per second to the right, and the initial velocity of the ball B is 12 meters per second to the left.
1. When the two balls hit each other, what will happen if it is a perfectly inelastic collision?
2. Compare the final velocity of the balls.
3. What can you say about the total momentum before and after the collision?
initial momentum to right = 7 * 6 - 2 * 12 = 18 kg m/s
So 18 will be the final momentum to the right as well
so if u is ball a and v is ball b
18 = 7 u + 2 v
now about kinetic energy
initial =(1/2) 7 (36) + (1/2)(2)(144) = 126+72 = 198 Joules
since we said inelastic final = initial so
(1/2) 7 u^2 + (1/2)(2) v^2 = 198
well v= (18-7u)/2
3.5 u^2 + (1/4)(324 - 252 u + 49 u^2) = 198
14 u^2 + 49 u^2 -252u -468 = 0
63 u^2 - 252 u - 468 = 0
https://www.mathsisfun.com/quadratic-equation-solver.html
u = 5.38 or -1.38
which?
if -1.38 then v = (18-7u)/2 = (18 + 9.66)/2 = 13.83 possible
if 5.38 then v = (18 -37.66)/2 =- 9.33 possible only if second ball passed right through the first one
so u = -1.38 and v = 13.83
Newton's very first said the momentum did not change. That is what we used.
Given: M1 = 7kg, V1 = 6m/s.
M2 = 2kg, V2 = -12m/s.
V3 = velocity of M1 and M2 after collision.
Momentum before = Momentum after.
M!*V1 + M2*V2 = M1*V3 + M2*V3.
7*6 + 2*(-12) = 7*V3 + 2*V3
9V3 = 18
V3 = 2 m/s.
physics maths
To answer these questions, we need to understand the principles of momentum and the concept of perfectly inelastic collisions.
1. In a perfectly inelastic collision, the two objects stick together after the collision and move as one. So, when the two balls collide, they will stick together and move as one object.
2. To find the final velocity of the balls after the collision, we need to use the conservation of momentum. The momentum of an object is given by the product of its mass and velocity. The total momentum before the collision is equal to the total momentum after the collision.
The initial momentum of ball A is calculated by multiplying its mass (7 kg) by its velocity (6 m/s to the right), resulting in a momentum of 42 kg*m/s to the right. The initial momentum of ball B is calculated by multiplying its mass (2 kg) by its velocity (12 m/s to the left), resulting in a momentum of 24 kg*m/s to the left.
Since momentum is conserved in a perfectly inelastic collision, the total momentum before the collision is equal to the total momentum after the collision. Therefore, the sum of the initial momenta (42 kg*m/s - 24 kg*m/s) equals the final momentum of the system.
The sum of the masses of the balls is 7 kg + 2 kg = 9 kg. Therefore, the final momentum of the balls (as they stick together) is (42 kg*m/s - 24 kg*m/s) = 18 kg*m/s to the right.
To find the final velocity, we divide the final momentum by the total mass of the system (9 kg). The final velocity is then 18 kg*m/s divided by 9 kg, resulting in a final velocity of 2 m/s to the right. Thus, the final velocity of the two balls together is 2 m/s to the right.
3. The principle of momentum conservation tells us that the total momentum before a collision is equal to the total momentum after the collision in an isolated system (no external forces). In this scenario, the total momentum before the collision is 42 kg*m/s to the right (from ball A) minus 24 kg*m/s to the left (from ball B). After the collision, the total momentum is 18 kg*m/s to the right (as the two balls stick together). Thus, the total momentum is conserved both before and after the collision.