"Cart 1 has a mass of 2.8 kg and is moving with a velocity of 6.6 m/s[right] along a frictionless track. Cart 2 has a mass of 3.2 kg and is moving at 2.5 m/s[left]. The carts collide in a head-on elastic collision cushioned by a spring with a spring constant of 3.0x10^4 N/m. Calculate the maximum compression of the spring."
Options are: 0.069, 0.041, 0.077, 0.096, 0.064
Ok so I'm having trouble with this question. I keep getting an answer that doesn't fit any options that are given...
I know I have to find the velocity of both carts at max. compression and got 4.413 as my answer:((m1*v1)+(m2*v2))/(m1+m2)
Then I tried calculating for max. compression using this equation:
((m1)(v1)^2 + (m2)(v2)^2 - (m1+m2)(4.413)^2)/k --- (all under the root)
and I got 0.0289 as an answer. What am I doing wrong?
Using conservation of energy (sume of intial KE goes into PE of spring), I get 0.930965108 m, which is not one of the answer either. PS. I disagree with both of the ways you worked it.
The way I solved it was the way the textbook recommended to solve it; this question just had different numbers than the textbook's. That's why I'm confused about how I didn't get the answer. Thanks anyway for your help.
To solve this problem, you need to use the principle of conservation of momentum and conservation of kinetic energy.
First, let's find the initial momentum of the system. The momentum of each cart is calculated by multiplying its mass by its velocity:
Momentum of cart 1 = (mass of cart 1) x (velocity of cart 1) = 2.8 kg x 6.6 m/s = 18.48 kg·m/s [right]
Momentum of cart 2 = (mass of cart 2) x (velocity of cart 2) = 3.2 kg x (-2.5 m/s) = -8 kg·m/s [left] (Note that the negative sign indicates the direction of the velocity)
The total initial momentum of the system is the sum of the individual momenta:
Total initial momentum = Momentum of cart 1 + Momentum of cart 2 = 18.48 kg·m/s - 8 kg·m/s = 10.48 kg·m/s [right]
Next, let's find the initial kinetic energy of the system. The kinetic energy of each cart is calculated by using the formula:
Kinetic energy = (1/2) x (mass of cart) x (velocity of cart)^2
Kinetic energy of cart 1 = (1/2) x 2.8 kg x (6.6 m/s)^2 = 68.904 J
Kinetic energy of cart 2 = (1/2) x 3.2 kg x (2.5 m/s)^2 = 20 J
The total initial kinetic energy of the system is the sum of the individual kinetic energies:
Total initial kinetic energy = Kinetic energy of cart 1 + Kinetic energy of cart 2 = 68.904 J + 20 J = 88.904 J
In an elastic collision, the total momentum and total kinetic energy of the system are conserved. Therefore, the total final momentum and total final kinetic energy will be equal to the initial values.
Now let's calculate the final velocity of cart 1 using the conservation of momentum:
Final momentum of cart 1 = Total initial momentum = 10.48 kg·m/s
Final momentum of cart 2 = Total initial momentum = 10.48 kg·m/s
Since the masses of the two carts are the same, the final velocities of the carts will be the same (but in opposite directions) due to conservation of momentum.
Therefore, final velocity of cart 1 = -final velocity of cart 2 = - (10.48 kg·m/s) / (2.8 kg + 3.2 kg)
Final velocity of cart 1 = - (10.48 kg·m/s) / 6 kg ≈ -1.747 m/s [left]
Now that we have the final velocities, we can calculate the maximum compression of the spring using the equation for the potential energy stored in a spring:
Potential energy stored in the spring = (1/2) x (spring constant) x (maximum compression)^2
At maximum compression, the total kinetic energy of the system is converted into potential energy stored in the spring.
Total final kinetic energy = Potential energy stored in the spring
(1/2) x (mass of cart 1) x (final velocity of cart 1)^2 + (1/2) x (mass of cart 2) x (final velocity of cart 2)^2 = (1/2) x (spring constant) x (maximum compression)^2
Plugging in the values:
(1/2) x 2.8 kg x (-1.747 m/s)^2 + (1/2) x 3.2 kg x (1.747 m/s)^2 = (1/2) x (3.0x10^4 N/m) x (maximum compression)^2
Simplifying and solving for the maximum compression:
1.216 J + 10.883 J = (1.5x10^4 N/m) x (maximum compression)^2
12.099 J = (1.5x10^4 N/m) x (maximum compression)^2
Dividing both sides by (1.5x10^4 N/m):
(maximum compression)^2 = (12.099 J) / (1.5x10^4 N/m)
(maximum compression)^2 ≈ 0.0008066
Taking the square root of both sides:
maximum compression ≈ 0.0284 m
So, the correct option from the provided list is 0.0284.