Two point charges, q1 and q2 are separated by a distance d. For convenience, we will say that q1 is at the origin (0,0) on a coordinate system, q2 is at (d,0) and a reference point p is directly above q2 at a distance h at point (d,h). q1 has value 1μ C and q2 has value 8μ C. Take d= 1.1m and h= 0.7m. Note k=9×109Nm2/C2 q1 and q2 both have mass 0.1 kg. If they are released, what will be the final speed of q1 (when the charges collide or when they are far enough apart that they no longer interact)? Give your answer in m/s but do not include the units in your response. Include three significant digits in your answer.
Question
Two point charges, q1 and q2 are separated by a distance d. For convenience, we will say that q1 is at the origin (0,0) on a coordinate system, q2 is at (d,0) and a reference point p is directly above q2 at a distance h at point (d,h). q1 has value 1μ C and q2 has value 8μ C. Take d= 1.1m and h= 0.7m.
Note k=9×109Nm2/C2 q1 and q2 both have mass 0.1 kg. If they are released, what will be the final speed of q1 (when the charges collide or when they are far enough apart that they no longer interact)? Give your answer in m/s but do not include the units in your response. Include three significant digits in your answer.
Solution
To solve this problem, we need to consider the conservation of energy and the conservation of momentum.
First, let's calculate the initial potential energy of the system. The potential energy (U) between two charges is given by the formula:
U = kq1q2/d
where k is Coulomb's constant (9x10^9 Nm^2/C^2), q1 and q2 are the charges, and d is the distance between them. Plugging in the given values:
U = 9x10^9 * 1x10^-6 * 8x10^-6 / 1.1 = 0.06545 Joules
Next, let's calculate the initial momentum of the system. Since the charges are initially at rest, the initial momentum is zero.
Now, let's consider the final state of the system. The charges will either collide or be far enough apart that they no longer interact. In either case, the final potential energy of the system will be zero.
By conservation of energy, the initial potential energy will be converted into kinetic energy. The kinetic energy (K) of an object is given by the formula:
K = 1/2 * m * v^2
where m is the mass of the object and v is its speed. Since the charges have the same mass, the final speed of q1 will be the same as the final speed of q2. Let's denote this speed as v.
By conservation of momentum, the final momentum of the system will be the same as the initial momentum, which is zero. This means that the momentum of q1 will be equal and opposite to the momentum of q2. The momentum (p) of an object is given by the formula:
p = m * v
Setting the momentum of q1 equal to the negative of the momentum of q2 gives:
0.1 * v = -0.1 * v
This equation is satisfied by v = 0, which means that the charges will not move. Therefore, the final speed of q1 is 0 m/s.
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