Two carts with masses of 4.3 kg and 3.0 kg
move toward each other on a frictionless track
with speeds of 5.7 m/s and 4.4 m/s, respectively. The carts stick together after colliding
head-on.
Find their final speed.
Answer in units of m/s
To find the final speed of the two carts after they collide and stick together, 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.
Momentum (p) is calculated by multiplying mass (m) by velocity (v). So the momentum before the collision for the first cart (m1) with mass of 4.3 kg and velocity of 5.7 m/s can be calculated as:
p1 = m1 * v1
= 4.3 kg * 5.7 m/s
= 24.51 kg·m/s
Similarly, the momentum before the collision for the second cart (m2) with mass of 3.0 kg and velocity of 4.4 m/s can be calculated as:
p2 = m2 * v2
= 3.0 kg * 4.4 m/s
= 13.2 kg·m/s
The total momentum before the collision (p_total) is the sum of the momenta of both carts:
p_total = p1 + p2
= 24.51 kg·m/s + 13.2 kg·m/s
= 37.71 kg·m/s
Since the carts stick together after the collision, they will have a combined mass (m_final) of the sum of their individual masses:
m_final = m1 + m2
= 4.3 kg + 3.0 kg
= 7.3 kg
Now, to find the final velocity (v_final), we divide the total momentum (p_total) by the combined mass (m_final):
v_final = p_total / m_final
= 37.71 kg·m/s / 7.3 kg
= 5.17 m/s
Therefore, the final speed of the two carts after the collision is 5.17 m/s.
To find the final velocity of the two carts after they stick together, we can apply the law of conservation of momentum.
The law of conservation of momentum states that the total momentum before a collision is equal to the total momentum after the collision, as long as no external forces are acting on the system.
The momentum (p) of an object is calculated by multiplying its mass (m) by its velocity (v): p = m * v.
Using this information, we can calculate the total momentum before the collision:
Total momentum before = (mass of cart 1 * velocity of cart 1) + (mass of cart 2 * velocity of cart 2)
In this case, the mass of cart 1 is 4.3 kg, the velocity of cart 1 is 5.7 m/s, the mass of cart 2 is 3.0 kg, and the velocity of cart 2 is -4.4 m/s (since it is moving in the opposite direction). Note that the negative sign indicates the opposite direction.
Total momentum before = (4.3 kg * 5.7 m/s) + (3.0 kg * -4.4 m/s)
Calculating this gives us:
Total momentum before = 24.51 kg*m/s + (-13.2 kg*m/s) = 11.31 kg*m/s
Since the carts stick together after the collision, their masses are combined into one. The mass of the combined carts will be the sum of the masses of cart 1 and cart 2:
Combined mass = mass of cart 1 + mass of cart 2 = 4.3 kg + 3.0 kg = 7.3 kg
Now, we can calculate the final velocity using the total momentum after the collision:
Total momentum after = combined mass of the carts * final velocity
Since the total momentum before and after the collision is the same, we can set up the equation:
Total momentum before = Total momentum after
11.31 kg*m/s = 7.3 kg * final velocity
Solving for the final velocity:
final velocity = 11.31 kg*m/s / 7.3 kg
final velocity ≈ 1.55 m/s
Therefore, the final speed of the combined carts is approximately 1.55 m/s.