a 4 kg object moving with a speed of 7.8 m/s collides with a 16 kg object moving with a velocity of 7.6 m/s in a direction of 22 degrees from the initial direction of motion the 4 kg object. what is the speed of the two objects after the collision if they remain stuck together
To solve this problem, 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.
The momentum (p) of an object is given by the formula:
p = m * v,
where:
- p is the momentum,
- m is the mass of the object, and
- v is the velocity of the object.
Before the collision:
The momentum of the 4 kg object can be calculated as:
p1 = m1 * v1 = 4 kg * 7.8 m/s = 31.2 kg·m/s.
To calculate the momentum of the 16 kg object, we need to find its velocity in the direction of motion of the 4 kg object. The component of the 16 kg object's velocity in this direction can be calculated as:
v1_16 = 7.6 m/s * cos(22°) ≈ 7.6 m/s * 0.927 ≈ 7.06 m/s.
Therefore, the momentum of the 16 kg object can be calculated as:
p2 = m2 * v1_16 = 16 kg * 7.06 m/s = 113 kg·m/s.
The total momentum before the collision is:
p_total_initial = p1 + p2 = 31.2 kg·m/s + 113 kg·m/s = 144.2 kg·m/s.
After the collision:
As the two objects stick together, their combined mass will be the sum of the individual masses:
m_total = m1 + m2 = 4 kg + 16 kg = 20 kg.
Let's assume the final velocity of the combined objects after the collision is v_final.
The momentum of the combined objects after the collision will be:
p_total_final = m_total * v_final.
According to the conservation of momentum principle, the total momentum before the collision should be equal to the total momentum after the collision:
p_total_initial = p_total_final.
So we can write the equation as:
p1 + p2 = m_total * v_final.
Substituting the given values, we have:
31.2 kg·m/s + 113 kg·m/s = 20 kg * v_final.
Simplifying the equation:
144.2 kg·m/s = 20 kg * v_final.
To find v_final, divide both sides of the equation by 20 kg:
v_final = 144.2 kg·m/s / 20 kg.
Therefore, the speed of the two objects after the collision if they remain stuck together is approximately:
v_final ≈ 7.21 m/s.