A 4.0 kg mass is moving along the x axis. The potential energy curve as a function of position is shown. The kinetic energy of the object at the origin is 12 J. The system is conservative. There is no friction.
(a) What will be the kinetic energy at 2.0 m along the +x axis?
(b) What will be the speed of the object at 6.0 m along the +x axis?
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To find the kinetic energy at a specific point along the x-axis, we first need to understand the relationship between potential energy and kinetic energy.
The total mechanical energy of the system, which includes the potential energy (PE) and kinetic energy (KE), remains constant in the absence of external forces. This is known as the law of conservation of mechanical energy.
In this case, since there is no friction and the system is conservative, we know that the total mechanical energy will be conserved.
The total mechanical energy (TME) is given by the sum of the potential energy (PE) and the kinetic energy (KE):
TME = PE + KE
Now, let's go through the steps to find the kinetic energy at 2.0 m and the speed at 6.0 m along the +x axis.
(a) To find the kinetic energy at 2.0 m:
1. Find the potential energy at the origin (x = 0 m).
Reading the potential energy curve, we see that the potential energy at the origin is zero.
PE(0) = 0 J
2. Use the conservation of mechanical energy to find the total mechanical energy (TME) at the origin (x = 0 m):
Since the kinetic energy at the origin is given as 12 J, we can use the equation:
TME(0) = PE(0) + KE(0)
TME(0) = 0 J + 12 J
TME(0) = 12 J
3. Use the conservation of mechanical energy to find the total mechanical energy (TME) at 2.0 m:
We can assume that the potential energy at 2.0 m is PE(2.0) J.
TME(2.0) = PE(2.0) + KE(2.0)
TME(2.0) = PE(2.0) + KE(0) (Since kinetic energy is given as 12 J at the origin)
TME(2.0) = PE(2.0) + 12 J
4. Substitute the values and solve for PE(2.0):
TME(2.0) = 12 J (TME is conserved)
PE(2.0) + 12 J = 12 J (Substitute TME(2.0) = 12 J)
PE(2.0) = 0 J
Therefore, the potential energy at 2.0 m is zero, which means all the energy is converted into kinetic energy.
KE(2.0) = TME(2.0) - PE(2.0)
KE(2.0) = 12 J - 0 J
KE(2.0) = 12 J
So, the kinetic energy at 2.0 m along the +x axis is 12 J.
(b) To find the speed at 6.0 m:
1. Find the potential energy at 6.0 m:
Read the potential energy value at 6.0 m on the potential energy curve.
PE(6.0) = <<potential energy at 6.0 m>>
2. Use the conservation of mechanical energy to find the total mechanical energy (TME) at 6.0 m:
TME(6.0) = PE(6.0) + KE(6.0)
3. Substitute the values and solve for KE(6.0):
TME(6.0) = <<total mechanical energy at 6.0 m>> (The value can be obtained from the conservation of mechanical energy)
PE(6.0) + KE(6.0) = <<total mechanical energy at 6.0 m>>
KE(6.0) = TME(6.0) - PE(6.0)
Therefore, the kinetic energy at 6.0 m can be calculated using the total mechanical energy and the potential energy at that point.
4. Once you have the kinetic energy, you can calculate the speed at 6.0 m using the equation:
KE(6.0) = (1/2)mv^2
Solve for v to find the speed at 6.0 m along the +x axis.
Note: To fully solve this problem, you need the values of the potential energy at 6.0 m and the total mechanical energy at 6.0 m. Please provide those values to get the specific answer.