A sphere of mass 500g moving with a velocity of 200 cm s-1collides centrally with another sphere of mass 100 g moving with a velocity of 100 cm s-1towards it. After the collision the two spheres stick together. Find the final velocities of the two spheres and the loss in kinetic energy of the system.
Plz answer fast
To find the final velocities of the two spheres and the loss in kinetic energy of the system, we can use the principle of conservation of momentum and conservation of kinetic energy.
1. Conservation of momentum:
The total momentum before the collision is equal to the total momentum after the collision.
Momentum before collision = Momentum after collision
Let's calculate the initial and final momenta separately:
Initial momentum:
- The mass of the first sphere is 500g = 0.5 kg
- The mass of the second sphere is 100g = 0.1 kg
- The velocity of the first sphere is 200 cm/s = 2 m/s (since 1 m = 100 cm)
- The velocity of the second sphere is -100 cm/s = -1 m/s (opposite direction)
Initial momentum = (mass of first sphere * velocity of first sphere) + (mass of second sphere * velocity of second sphere)
= (0.5 kg * 2 m/s) + (0.1 kg * -1 m/s)
= 1 kg m/s - 0.1 kg m/s
= 0.9 kg m/s
Final momentum:
- After the collision, the two spheres stick together and move with a common velocity.
- Let's assume this common velocity as v.
Final momentum = (mass of the combined spheres * common velocity)
= (0.6 kg * v)
According to conservation of momentum,
Initial momentum = Final momentum
0.9 kg m/s = 0.6 kg * v
Solving for v, we get:
v = 0.9 kg m/s / 0.6 kg
v = 1.5 m/s
So, the common final velocity of the two spheres is 1.5 m/s.
2. Conservation of kinetic energy:
The total kinetic energy before the collision is equal to the total kinetic energy after the collision.
Kinetic energy before collision = Kinetic energy after collision
The kinetic energy of an object is given by the formula:
Kinetic energy = (1/2) * mass * (velocity)^2
Let's calculate the initial and final kinetic energies separately:
Initial kinetic energy:
- The mass of the first sphere is 500g = 0.5 kg
- The velocity of the first sphere is 200 cm/s = 2 m/s
Initial kinetic energy = (1/2) * (mass of first sphere) * (velocity of first sphere)^2
= (1/2) * 0.5 kg * (2 m/s)^2
= (1/2) * 0.5 kg * 4 m^2/s^2
= 1 J
Final kinetic energy:
- After the collision, the two spheres stick together and move with a common velocity of 1.5 m/s (as we calculated earlier).
- The combined mass of the two spheres is 0.6 kg.
Final kinetic energy = (1/2) * (mass of combined spheres) * (common velocity)^2
= (1/2) * 0.6 kg * (1.5 m/s)^2
= (1/2) * 0.6 kg * 2.25 m^2/s^2
= 0.675 J
The loss in kinetic energy of the system is the difference between the initial kinetic energy and the final kinetic energy:
Loss in kinetic energy = Initial kinetic energy - Final kinetic energy
= 1 J - 0.675 J
= 0.325 J
So, the final velocities of the two spheres are 1.5 m/s (common velocity), and the loss in kinetic energy of the system is 0.325 J.
To solve this problem, we need to apply the laws of conservation of momentum and conservation of kinetic energy.
First, let's find the initial momentum and kinetic energy of the system before the collision.
The initial momentum (p) of an object is given by the product of its mass (m) and velocity (v). So, the initial momentum of the first sphere (m1) is 500g (0.5kg) × 200 cm/s and the initial momentum of the second sphere (m2) is 100g (0.1kg) × (-100 cm/s) since it is moving towards the first sphere.
Initial momentum of the system (before collision) = m1 × v1 + m2 × v2
= (0.5kg × 200 cm/s) + (0.1kg × (-100 cm/s))
= 100 kg cm/s - 10 kg cm/s
= 90 kg cm/s
Now, let's find the initial kinetic energy (KE) of the system:
Kinetic energy (KE) = 0.5 × mass × velocity^2
Initial KE of the system (before collision) = 0.5 × (m1 × v1^2 + m2 × v2^2)
= 0.5 × (0.5kg × (200 cm/s)^2 + 0.1kg × (100 cm/s)^2)
= 0.5 × (0.5kg × 40000 cm^2/s^2 + 0.1kg × 10000 cm^2/s^2)
= 0.5 × (20000 kg cm^2/s^2 + 1000 kg cm^2/s^2)
= 0.5 × 21000 kg cm^2/s^2
= 10500 kg cm^2/s^2
Now, let's find the final velocity (v') of the two spheres after the collision, assuming they stick together.
According to the law of conservation of momentum, the total momentum of a closed system before and after the collision remains constant.
Final momentum (p') of the system (after collision) = m' × v'
Since the two spheres stick together, their masses add up:
m' = m1 + m2 = 0.5kg + 0.1kg = 0.6kg
Moreover, the final velocity (v') is the same for both spheres after sticking together.
Therefore, we can rewrite the conservation of momentum equation as:
(m1 × v1) + (m2 × v2) = (m' × v')
(0.5kg × 200 cm/s) + (0.1kg × (-100 cm/s)) = (0.6kg × v')
100 kg cm/s - 10 kg cm/s = 0.6kg × v'
90 kg cm/s = 0.6kg × v'
v' = 90 kg cm/s / 0.6kg
v' = 150 cm/s
So, the final velocity of the two spheres after the collision is 150 cm/s.
To find the loss in kinetic energy of the system, we subtract the final kinetic energy from the initial kinetic energy:
Loss in kinetic energy = Initial KE - Final KE
= 10500 kg cm^2/s^2 - 0.5 × (0.6kg × (150 cm/s)^2)
= 10500 kg cm^2/s^2 - 0.5 × (0.6kg × 22500 cm^2/s^2)
= 10500 kg cm^2/s^2 - 0.5 × 13500 kg cm^2/s^2
= 10500 kg cm^2/s^2 - 6750 kg cm^2/s^2
= 3750 kg cm^2/s^2
Therefore, the loss in kinetic energy of the system is 3750 kg cm^2/s^2.
use meters and kilograms or I get confused
original momentum = 0.5 * 0.2 - 0.1 * 0.1= 0.09 kg m/s
final momentum = 0.6*v
final momentum = original so
0.6 v = 0.09
v = 0.15 m/s
original ke = (1/2)[0.5*(0.2)^2 + 0.1(0.1)^2] = .5[.02+.001]
= 0.0105 Joules
final ke =.5[0.6 *(0.15)^2] = 0.00675 Joules
subtract them
Given: M1 = 0.5 kg, V1 = 2 m/s.
M2 = 0.1kg, V2 = 1m/s.
V3 = Velocity 0f M1 and M2 after collision.
Momentum bef0re = Momentum after.
M1*V1+M2*V2 = M1*V3+M2*V3
0.5*2+0.1*(-1) = 0.5V3+0.1V3
0.6V3 = 0.9
V3 = 1.5 m/s.
KEb = 0.5M1*V1^2+0.5M2*V2^2 = 0.25*2^2+0.05*(-1)^2 = 1.05 Joules.
KEa = 0.5M1*V3^2+0.5M2*V3^2 = 0.25*1.5^2+0.05*1.5^2 = 0.675 Joules.
KE lost = KEb-KEa =