A 2.0-kg object moving with a velocity of 5.0 m/s in the positive x
direction strikes and sticks to a 3.0-kg object moving with a speed of 2.0 m/s in the same direction. How much kinetic energy is lost in this collision?
To determine the amount of kinetic energy lost in this collision, we first need to calculate the initial kinetic energy before the collision and the final kinetic energy after the collision.
The initial kinetic energy (KEi) can be calculated using the formula:
KEi = (1/2) * m1 * v1^2 + (1/2) * m2 * v2^2
Where,
m1 and v1 are the mass and velocity of the first object, respectively,
m2 and v2 are the mass and velocity of the second object, respectively.
In this case, the mass of the first object (m1) is 2.0 kg, and its velocity (v1) is 5.0 m/s. The mass of the second object (m2) is 3.0 kg, and its velocity (v2) is 2.0 m/s. Plugging these values into the equation, we get:
KEi = (1/2) * 2.0 kg * (5.0 m/s)^2 + (1/2) * 3.0 kg * (2.0 m/s)^2
= 100 J + 6 J
= 106 J
Now, let's calculate the final kinetic energy (KEf) after the collision. Since the two objects stick together, their combined mass will be the sum of their individual masses. Let's denote this mass as m_f, and the final velocity of the combined object as v_f.
The equation to calculate the final kinetic energy is:
KEf = (1/2) * m_f * v_f^2
Initially, the two objects were moving in the same direction, so they will have the same final velocity (v_f) after sticking together. Therefore, we have:
m_f = m1 + m2 = 2.0 kg + 3.0 kg = 5.0 kg
Plugging this into the equation, we get:
KEf = (1/2) * 5.0 kg * v_f^2
The kinetic energy lost in the collision can be calculated by subtracting the final kinetic energy from the initial kinetic energy:
Kinetic energy lost = KEi - KEf
So, to calculate the kinetic energy lost, we need to find the final velocity (v_f) of the combined object after the collision. To do this, we can use the principle of conservation of momentum.
According to the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision. Mathematically, this can be expressed as:
(m1 * v1) + (m2 * v2) = mf * vf
Plugging in the values, we have:
(2.0 kg * 5.0 m/s) + (3.0 kg * 2.0 m/s) = 5.0 kg * vf
10 kg m/s + 6 kg m/s = 5 kg * vf
16 kg m/s = 5 kg * vf
vf = 16 kg m/s / 5 kg
vf = 3.2 m/s
Now that we have the final velocity (vf), we can calculate the final kinetic energy (KEf):
KEf = (1/2) * 5.0 kg * (3.2 m/s)^2
= (1/2) * 5.0 kg * 10.24 m^2/s^2
= 25.6 J
Finally, we can calculate the kinetic energy lost:
Kinetic energy lost = KEi - KEf
= 106 J - 25.6 J
= 80.4 J
Therefore, the kinetic energy lost in this collision is 80.4 Joules.
To find out how much kinetic energy is lost in this collision, we need to calculate the initial kinetic energy and the final kinetic energy, and then find the difference between them.
The initial kinetic energy of an object can be calculated using the formula:
Kinetic Energy = (1/2) * mass * velocity^2
For the first object with a mass of 2.0 kg and velocity of 5.0 m/s, the initial kinetic energy is:
Kinetic Energy1 = (1/2) * 2.0 kg * (5.0 m/s)^2
= (1/2) * 2.0 kg * 25.0 m^2/s^2
= 25.0 J
For the second object with a mass of 3.0 kg and velocity of 2.0 m/s, the initial kinetic energy is:
Kinetic Energy2 = (1/2) * 3.0 kg * (2.0 m/s)^2
= (1/2) * 3.0 kg * 4.0 m^2/s^2
= 6.0 J
Total initial kinetic energy = Kinetic Energy1 + Kinetic Energy2
= 25.0 J + 6.0 J
= 31.0 J
After the collision, the two objects stick together and move together as a single object.
The final kinetic energy can be calculated using the new combined mass and their combined velocity.
Total mass after collision = mass1 + mass2
= 2.0 kg + 3.0 kg
= 5.0 kg
Total velocity after collision = (mass1 * velocity1 + mass2 * velocity2) / total mass
= (2.0 kg * 5.0 m/s + 3.0 kg * 2.0 m/s) / 5.0 kg
= (10.0 kgm/s + 6.0 kgm/s) / 5.0 kg
= 16.0 kgm/s / 5.0 kg
= 3.2 m/s
The final kinetic energy is given by:
Final Kinetic Energy = (1/2) * total mass * (total velocity)^2
= (1/2) * 5.0 kg * (3.2 m/s)^2
= (1/2) * 5.0 kg * 10.24 m^2/s^2
= 25.6 J
The kinetic energy lost in this collision is the difference between the initial and final kinetic energy:
Kinetic Energy lost = Total initial kinetic energy - Final kinetic energy
= 31.0 J - 25.6 J
= 5.4 J
Therefore, the kinetic energy lost in this collision is 5.4 Joules.
m1•v1+m2•v2 =(m1+m2)•u
u= (m1•v1+m2•v2)/ (m1+m2) =
=(2•5+3•2)/(5+3) =2 m/s
m1•v1²/2 + m2•v2²/2 –(m1+m2) •u²/2 =
= 2•25/2 +3•4/2 –8•4/2 =15 J