In an elastic collision, a 1.0 kg ball moving at 1.0 m/s collides with a 2.0 kg ball moving at −2.0 m/s. The 2.0 kg ball transfers all of its momentum to the 1.0 kg ball.
What velocity does the 1.0 kg ball have after the collision?
What is the initial kinetic energy of the system?
What is the final kinetic energy of the system?
The 1.0 kg ball has a velocity of 3.0 m/s after the collision.
The initial kinetic energy of the system is 4.0 J.
The final kinetic energy of the system is also 4.0 J.
Well, that sounds like a bouncy situation! Let's see what we can do with those balls.
To find the velocity of the 1.0 kg ball after the collision, we can use the principle of conservation of momentum. Since momentum is conserved, we can equate the momentum before the collision to the momentum after the collision.
Now, the momentum before the collision is:
Initial momentum = (mass 1 * velocity 1) + (mass 2 * velocity 2)
= (1.0 kg * 1.0 m/s) + (2.0 kg * (-2.0 m/s))
And since the 2.0 kg ball transfers all its momentum to the 1.0 kg ball,
Final momentum = (mass 1 * final velocity)
= 1.0 kg * final velocity
Setting the initial momentum equal to the final momentum, we can solve for the final velocity:
(1.0 kg * 1.0 m/s) + (2.0 kg * (-2.0 m/s)) = 1.0 kg * final velocity
Solving the equation, we find that the final velocity is approximately -1.0 m/s. So, the 1.0 kg ball has a velocity of -1.0 m/s after the collision. It's going the opposite way!
As for the initial kinetic energy of the system, we can calculate it using the formula:
Initial kinetic energy = 1/2 * (mass 1 * velocity 1^2) + 1/2 * (mass 2 * velocity 2^2)
Plugging in the values, we get:
Initial kinetic energy = 1/2 * (1.0 kg * (1.0 m/s)^2) + 1/2 * (2.0 kg * (-2.0 m/s)^2)
Calculating that, we find that the initial kinetic energy of the system is 5.0 Joules.
Now, for the final kinetic energy of the system after the collision, we can use the formula:
Final kinetic energy = 1/2 * (mass 1 * final velocity^2)
Plugging in the values, we get:
Final kinetic energy = 1/2 * (1.0 kg * (-1.0 m/s)^2)
Calculating that, we find that the final kinetic energy of the system is 0.5 Joules.
So, the 1.0 kg ball is going at -1.0 m/s after the collision, the initial kinetic energy was 5.0 Joules, and the final kinetic energy is 0.5 Joules. Talk about an energetic transformation!
To solve this problem, we can use the conservation of momentum and kinetic energy.
Step 1: Calculate the initial momentum of the system.
The momentum of an object is calculated by multiplying its mass by its velocity. Let's denote the mass of the 1.0 kg ball as m1 and the mass of the 2.0 kg ball as m2.
Initial momentum of the system = momentum of ball 1 + momentum of ball 2
= m1 * v1 + m2 * v2
Given:
m1 = 1.0 kg
v1 = 1.0 m/s
m2 = 2.0 kg
v2 = -2.0 m/s
Plugging in the values, we have:
Initial momentum of the system = (1.0 kg * 1.0 m/s) + (2.0 kg * -2.0 m/s)
Step 2: Calculate the final momentum of the system.
Since the 2.0 kg ball transfers all of its momentum to the 1.0 kg ball, the final momentum of the system will be equal to the momentum of the 1.0 kg ball after the collision.
Final momentum of the system = m * v'
where m is the mass of the 1.0 kg ball and v' is its velocity after the collision.
Step 3: Equate initial momentum to final momentum.
Initial momentum of the system = Final momentum of the system
m1 * v1 + m2 * v2 = m * v'
Step 4: Calculate the velocity of the 1.0 kg ball after the collision.
Rearranging the equation from step 3, we get:
v' = (m1 * v1 + m2 * v2) / m1
Plugging in the given values, we have:
v' = (1.0 kg * 1.0 m/s + 2.0 kg * -2.0 m/s) / 1.0 kg
Simplifying the expression, we get:
v' = (-3.0 kg m/s) / 1.0 kg
Therefore, the velocity of the 1.0 kg ball after the collision is -3.0 m/s.
Step 5: Calculate the initial kinetic energy of the system.
The kinetic energy of an object is given by 1/2 * mass * velocity^2.
Initial kinetic energy of the system = Kinetic energy of ball 1 + Kinetic energy of ball 2
= (1/2 * m1 * v1^2) + (1/2 * m2 * v2^2)
Plugging in the given values, we have:
Initial kinetic energy of the system = (1/2 * 1.0 kg * (1.0 m/s)^2) + (1/2 * 2.0 kg * (-2.0 m/s)^2)
Simplifying the expression, we get:
Initial kinetic energy of the system = 1/2 Joule + 8 Joule = 8.5 Joules.
Step 6: Calculate the final kinetic energy of the system.
Since the 2.0 kg ball transfers all of its momentum to the 1.0 kg ball, the final kinetic energy of the system will be the kinetic energy of the 1.0 kg ball after the collision.
Final kinetic energy of the system = 1/2 * m * v'^2
Plugging in the given values, we have:
Final kinetic energy of the system = 1/2 * 1.0 kg * (-3.0 m/s)^2
Simplifying the expression, we get:
Final kinetic energy of the system = 4.5 Joules.
Therefore, the velocity of the 1.0 kg ball after the collision is -3.0 m/s, the initial kinetic energy of the system is 8.5 Joules, and the final kinetic energy of the system is 4.5 Joules.
To find the velocity of the 1.0 kg ball after the collision, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision should be equal to the total momentum after the collision.
The formula for momentum is:
Momentum = mass × velocity
For the 1.0 kg ball, its initial momentum is given by:
Momentum1_initial = mass1 × velocity1_initial
Since the 1.0 kg ball is moving at 1.0 m/s initially, its initial momentum is:
Momentum1_initial = 1.0 kg × 1.0 m/s = 1.0 kg·m/s
The 2.0 kg ball transfers all of its momentum to the 1.0 kg ball, meaning its final momentum is completely transferred to the 1.0 kg ball. Therefore, we can say that the final momentum of the system (1.0 kg ball) is the same as the initial momentum of the 2.0 kg ball.
So, Momentum1_final = Momentum2_initial
The initial momentum of the 2.0 kg ball can be calculated as:
Momentum2_initial = mass2 × velocity2_initial
Since the 2.0 kg ball is moving at −2.0 m/s initially, its initial momentum is:
Momentum2_initial = 2.0 kg × (-2.0 m/s) = -4.0 kg·m/s
So, we have:
Momentum1_final = Momentum2_initial
1.0 kg·m/s = -4.0 kg·m/s
To find the velocity (velocity1_final) of the 1.0 kg ball after the collision, we divide its final momentum by its mass:
velocity1_final = Momentum1_final / mass1
Substituting the values we have:
velocity1_final = 1.0 kg·m/s / 1.0 kg
velocity1_final = 1.0 m/s
Therefore, the velocity of the 1.0 kg ball after the collision is 1.0 m/s.
Now, let's move on to the initial and final kinetic energy of the system.
The initial kinetic energy of the system is the sum of the individual kinetic energies of the two balls before the collision.
The formula for kinetic energy is:
Kinetic Energy = 0.5 × mass × velocity^2
For the 1.0 kg ball, its initial kinetic energy is:
Kinetic Energy1_initial = 0.5 × mass1 × velocity1_initial^2
Kinetic Energy1_initial = 0.5 × 1.0 kg × (1.0 m/s)^2
Kinetic Energy1_initial = 0.5 × 1.0 kg × 1.0 m^2/s^2
Kinetic Energy1_initial = 0.5 J (joules)
For the 2.0 kg ball, its initial kinetic energy is:
Kinetic Energy2_initial = 0.5 × mass2 × velocity2_initial^2
Kinetic Energy2_initial = 0.5 × 2.0 kg × (-2.0 m/s)^2
Kinetic Energy2_initial = 0.5 × 2.0 kg × 4.0 m^2/s^2
Kinetic Energy2_initial = 8.0 J (joules)
So, the initial kinetic energy of the system is:
Initial Kinetic Energy = Kinetic Energy1_initial + Kinetic Energy2_initial
Initial Kinetic Energy = 0.5 J + 8.0 J
Initial Kinetic Energy = 8.5 J (joules)
Finally, to find the final kinetic energy of the system, we calculate the kinetic energy of the 1.0 kg ball after the collision:
Kinetic Energy1_final = 0.5 × mass1 × velocity1_final^2
Kinetic Energy1_final = 0.5 × 1.0 kg × (1.0 m/s)^2
Kinetic Energy1_final = 0.5 × 1.0 kg × 1.0 m^2/s^2
Kinetic Energy1_final = 0.5 J (joules)
Since the 2.0 kg ball transfers all of its momentum to the 1.0 kg ball, we can say that its final velocity is zero. Therefore, its final kinetic energy is also zero.
So, the final kinetic energy of the system is:
Final Kinetic Energy = Kinetic Energy1_final + Kinetic Energy2_final
Final Kinetic Energy = 0.5 J + 0 J
Final Kinetic Energy = 0.5 J (joules)
Therefore, the final kinetic energy of the system is 0.5 Joules.