A ball of mass m(1) = 0.250 kg and initial v(1) = 5.00 m/s collides head-on with a ball of mass m(2) = 0.800 kg that is initially at rest. What are the velocities of the balls after the collision if they stick together?
To solve this problem, we can use the principle of conservation of linear momentum. According to this principle, the total linear momentum before the collision is equal to the total linear momentum after the collision.
Initially, only the first ball is moving, so the total initial momentum is given by:
p_initial = m1 * v1
where m1 is the mass of the first ball (0.250 kg) and v1 is its initial velocity (5.00 m/s).
After the collision, the two balls stick together, so they will have the same final velocity. Let's call this final velocity v_f.
The total final momentum is given by:
p_final = (m1 + m2) * v_f
where m2 is the mass of the second ball (0.800 kg).
According to the principle of conservation of linear momentum, we can equate the initial and final momenta:
p_initial = p_final
m1 * v1 = (m1 + m2) * v_f
Now we can solve for v_f:
v_f = (m1 * v1) / (m1 + m2)
= (0.250 kg * 5.00 m/s) / (0.250 kg + 0.800 kg)
= (1.25 kg m/s) / (1.05 kg)
≈ 1.19 m/s
Therefore, the final velocity of the balls after the collision, when they stick together, is approximately 1.19 m/s.
To solve this problem, we can make use of the principle of conservation of momentum. According to this principle, the total momentum of an isolated system remains constant before and after a collision.
The momentum of an object can be calculated using the formula:
Momentum (p) = mass (m) × velocity (v)
Given that ball 1 has a mass of 0.250 kg and an initial velocity of 5.00 m/s, its initial momentum can be calculated as:
p(1i) = m(1) × v(1)
= 0.250 kg × 5.00 m/s
= 1.25 kg·m/s
Since ball 2 is initially at rest, its initial momentum is:
p(2i) = m(2) × v(2)
= 0.800 kg × 0 m/s
= 0 kg·m/s
The total initial momentum of the system is the sum of the individual momenta:
p(i) = p(1i) + p(2i)
= 1.25 kg·m/s + 0 kg·m/s
= 1.25 kg·m/s
Since the balls stick together after the collision, they will move with a common final velocity (v(f)). Let's assume v(f) is the common final velocity of both balls.
The final momentum (p(f)) of the system can be calculated as:
p(f) = (m(1) + m(2)) × v(f)
Using the principle of conservation of momentum, we can equate the initial and final momenta:
p(i) = p(f)
Substituting the values, we have:
1.25 kg·m/s = (0.250 kg + 0.800 kg) × v(f)
Now we can solve for v(f):
v(f) = 1.25 kg·m/s / (0.250 kg + 0.800 kg)
= 1.25 kg·m/s / 1.050 kg
≈ 1.19 m/s
Therefore, after the collision, the balls will move together with a final velocity of approximately 1.19 m/s.
m₁v₁=(m₁+m₂)u
u= m₁v₁/(m₁+m₂)