An air-track cart with mass m=0.20kg and speed v0=1.5m/s approaches two other carts that are at rest and have masses 2m and 3m, as indicated in (Figure 1) . The carts have bumpers that make all the collisions elastic.
Find the final speed of cart 1, assuming the air track extends indefinitely in either direction.
Find the final speed of cart 2, assuming the air track extends indefinitely in either direction.
Find the final speed of cart 3, assuming the air track extends indefinitely in either direction.
To find the final speeds of the carts after the collision, we can apply the principles of conservation of momentum and kinetic energy.
Conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision. The formula for momentum is:
p = mv
Where p is momentum, m is mass, and v is velocity.
Conservation of kinetic energy states that the total kinetic energy before the collision is equal to the total kinetic energy after the collision. The formula for kinetic energy is:
KE = (1/2)mv^2
Where KE is kinetic energy, m is mass, and v is velocity.
Now, let's solve each part of the question:
Part 1: Final speed of cart 1
Cart 1 is approaching two other carts, which are at rest. Let's label the velocities of cart 1, 2, and 3 as v1, v2, and v3 respectively.
Using conservation of momentum:
Momentum before collision = Momentum after collision
m * v0 = m * v1 + 2m * v2 + 3m * v3
Simplifying the equation:
v0 = v1 + 2v2 + 3v3
Using conservation of kinetic energy:
Kinetic energy before collision = Kinetic energy after collision
(1/2) * m * v0^2 = (1/2) * m * v1^2 + (1/2) * 2m * v2^2 + (1/2) * 3m * v3^2
Simplifying the equation:
v0^2 = v1^2 + 2v2^2 + 3v3^2
Now, we have two equations:
v0 = v1 + 2v2 + 3v3
v0^2 = v1^2 + 2v2^2 + 3v3^2
Solve these equations simultaneously to find the values of v1, v2, and v3.
Part 2: Final speed of cart 2
Since cart 2 is at rest initially, its velocity will not contribute to the initial momentum and kinetic energy. Therefore, after the collision, its velocity will be v2.
Part 3: Final speed of cart 3
Since cart 3 is also at rest initially, its velocity will not contribute to the initial momentum and kinetic energy. Therefore, after the collision, its velocity will be v3.
By solving the equations using conservation of momentum and kinetic energy, you can find the values of v1, v2, and v3, which will give you the final speeds of cart 1, cart 2, and cart 3 respectively.
To find the final speeds of the three carts after the elastic collisions, we can use the conservation of momentum and kinetic energy.
The conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision. Mathematically, this can be expressed as:
m1v1 + m2v2 + m3v3 = m1vf1 + m2vf2 + m3vf3
where m1, m2, and m3 are the masses of carts 1, 2, and 3 respectively, v1, v2, and v3 are their initial velocities, and vf1, vf2, and vf3 are their final velocities.
Similarly, the conservation of kinetic energy states that the total kinetic energy before the collision is equal to the total kinetic energy after the collision. Mathematically, this can be expressed as:
0.5*m1*(v1^2) + 0.5*m2*(v2^2) + 0.5*m3*(v3^2) = 0.5*m1*(vf1^2) + 0.5*m2*(vf2^2) + 0.5*m3*(vf3^2)
For the given problem, we have:
m1 = 0.20 kg
m2 = 2m = 0.20*2 = 0.40 kg
m3 = 3m = 0.20*3 = 0.60 kg
v1 = 1.5 m/s
v2 = 0 m/s (cart 2 is at rest)
v3 = 0 m/s (cart 3 is at rest)
Let's now calculate the final speeds of the carts one by one.
Step 1: Calculate the initial and final momenta for each cart.
Initial momentum:
p1_initial = m1 * v1 = 0.20 kg * 1.5 m/s = 0.30 kg*m/s
p2_initial = m2 * v2 = 0.40 kg * 0 m/s = 0 kg*m/s (cart 2 is at rest)
p3_initial = m3 * v3 = 0.60 kg * 0 m/s = 0 kg*m/s (cart 3 is at rest)
Final momentum:
As momentum is conserved, the final momentum of each cart remains the same as the initial momentum.
p1_final = p1_initial = 0.30 kg*m/s
p2_final = p2_initial = 0 kg*m/s
p3_final = p3_initial = 0 kg*m/s
Step 2: Use the conservation of kinetic energy to calculate the final speeds of each cart.
Kinetic energy before collision:
KE_initial = 0.5*m1*(v1^2) + 0.5*m2*(v2^2) + 0.5*m3*(v3^2) = 0.5*0.20 kg*(1.5^2) + 0.5*0.40 kg*(0^2) + 0.5*0.60 kg*(0^2) = 0.225 J
Kinetic energy after collision:
KE_final = 0.5*m1*(vf1^2) + 0.5*m2*(vf2^2) + 0.5*m3*(vf3^2)
Since the carts have bumpers that make all the collisions elastic, the total kinetic energy remains the same.
KE_final = KE_initial = 0.225 J
Now, we can solve for the final speeds of each cart.
For cart 1:
KE_final = 0.5*m1*(vf1^2)
0.225 J = 0.5*0.20 kg*(vf1^2)
vf1^2 = (0.225 J) / (0.5*0.20 kg)
vf1^2 = 2.25 m^2/s^2
vf1 = √(2.25 m^2/s^2) = 1.5 m/s
For cart 2:
KE_final = 0.5*m2*(vf2^2)
0.225 J = 0.5*0.40 kg*(vf2^2)
vf2^2 = (0.225 J) / (0.5*0.40 kg)
vf2^2 = 1.125 m^2/s^2
vf2 = √(1.125 m^2/s^2) ≈ 1.06 m/s
For cart 3:
KE_final = 0.5*m3*(vf3^2)
0.225 J = 0.5*0.60 kg*(vf3^2)
vf3^2 = (0.225 J) / (0.5*0.60 kg)
vf3^2 = 0.75 m^2/s^2
vf3 = √(0.75 m^2/s^2) ≈ 0.87 m/s
Therefore, the final speeds of the carts are:
vf1 = 1.5 m/s
vf2 ≈ 1.06 m/s
vf3 ≈ 0.87 m/s