A steel ball A of mass 20.0 kg moving with a speed of 2.0 ms−1
collides with another
ball B of mass 10.0 kg which is initially at rest. After the collision A moves off with a
speed of 1.0 ms−1
at an angle of 30º with its original direction of motion. Determine the
final velocity of B.
To determine the final velocity of ball B after the collision, we need to apply the principles of conservation of momentum and conservation of kinetic energy.
1. Conservation of momentum:
According to the conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.
The formula for momentum is given by:
momentum = mass × velocity
Before the collision:
The momentum of ball A = mass of A × velocity of A
The momentum of ball B = mass of B × velocity of B
After the collision:
The momentum of ball A = mass of A × final velocity of A
The momentum of ball B = mass of B × final velocity of B
Using the conservation of momentum, we have:
momentum before collision = momentum after collision
(mass of A × velocity of A) + (mass of B × velocity of B) = (mass of A × final velocity of A) + (mass of B × final velocity of B)
2. Conservation of kinetic energy:
According to the conservation of kinetic energy, the total kinetic energy before the collision is equal to the total kinetic energy after the collision.
The formula for kinetic energy is given by:
kinetic energy = 0.5 × mass × (velocity)^2
Before the collision:
The kinetic energy of ball A = 0.5 × mass of A × (velocity of A)^2
The kinetic energy of ball B = 0.5 × mass of B × (velocity of B)^2
After the collision:
The kinetic energy of ball A = 0.5 × mass of A × (final velocity of A)^2
The kinetic energy of ball B = 0.5 × mass of B × (final velocity of B)^2
Using the conservation of kinetic energy, we have:
kinetic energy before collision = kinetic energy after collision
(0.5 × mass of A × (velocity of A)^2) + (0.5 × mass of B × (velocity of B)^2) = (0.5 × mass of A × (final velocity of A)^2) + (0.5 × mass of B × (final velocity of B)^2)
Now we can substitute the given values into the equations:
mass of A = 20.0 kg
mass of B = 10.0 kg
velocity of A = 2.0 m/s
final velocity of A = 1.0 m/s
angle = 30º (with respect to the original direction)
Solving these equations will allow us to find the final velocity of ball B after the collision.
To determine the final velocity of ball B, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
We can calculate the total momentum before the collision using the formula:
Total momentum before collision = (mass of ball A) × (velocity of ball A) + (mass of ball B) × (velocity of ball B)
Given:
Mass of ball A (mA) = 20.0 kg
Mass of ball B (mB) = 10.0 kg
Velocity of ball A (vA) = 2.0 m/s
Velocity of ball B (vB) = 0 m/s (since it is initially at rest)
Total momentum before collision = (20.0 kg) × (2.0 m/s) + (10.0 kg) × (0 m/s)
= 40.0 kg·m/s
Now, let's determine the total momentum after the collision. Ball A moves off with a speed of 1.0 m/s at an angle of 30º with its original direction of motion. To find the final velocity of ball B, we need to decompose the momentum of ball A into horizontal and vertical components.
Horizontal component of momentum (pAx) = (mass of ball A) × (final velocity of ball A) × (cosine of the angle)
= (20.0 kg) × (1.0 m/s) × (cos 30º)
Vertical component of momentum (pAy) = (mass of ball A) × (final velocity of ball A) × (sine of the angle)
= (20.0 kg) × (1.0 m/s) × (sin 30º)
Since ball B is initially at rest, its momentum in the horizontal direction will be zero.
Total momentum after collision = pAx(horizontal component of A) + pAy(vertical component of A) + pB (momentum of ball B)
Since the total momentum before the collision is equal to the total momentum after the collision, we have:
Total momentum before collision = Total momentum after collision
40.0 kg·m/s = pAx + pAy + pB
Since the horizontal component of ball A's momentum is zero, we can simplify the equation:
40.0 kg·m/s = pAy + pB
Now, let's plug in the values we know:
pAy = (20.0 kg) × (1.0 m/s) × (sin 30º) = 10.0 kg·m/s
pB = final momentum of ball B (which we are trying to determine)
40.0 kg·m/s = 10.0 kg·m/s + pB
To solve for pB, we subtract 10.0 kg·m/s from both sides of the equation:
40.0 kg·m/s - 10.0 kg·m/s = pB
pB = 30.0 kg·m/s
Therefore, the final momentum of ball B is 30.0 kg·m/s.
The final momentum of ball B is equal to its mass multiplied by its final velocity:
pB = (mass of ball B) × (final velocity of ball B)
30.0 kg·m/s = (10.0 kg) × (final velocity of ball B)
To find the final velocity of ball B, we can rearrange the equation:
(final velocity of ball B) = 30.0 kg·m/s / 10.0 kg
(final velocity of ball B) = 3.0 m/s
Therefore, the final velocity of ball B is 3.0 m/s.
assign the initial path of A as either the x or y direction
find the initial momentum of A in the x and y directions
... then, find the final momentum of A in the x and y directions
... the difference, in x and in y, is the momentum imparted to B
use B's x and y momenta to find its final velocity