A charge of -2.73 μC is fixed in place. From a horizontal distance of 0.0465 m, a particle of mass 9.50 x 10-3 kg and charge -9.56 μC is fired with an initial speed of 97.2 m/s directly toward the fixed charge. What is the distance of closest approach?
To find the distance of closest approach, we can use the conservation of mechanical energy before and after the collision.
1. Calculate the electrostatic potential energy (PE) at the closest approach:
- The electric potential energy between the two charges can be calculated using the formula:
PE = k * (q1 * q2) / r
where k is the Coulomb's constant (8.99 x 10^9 N m^2/C^2), q1 and q2 are the charges, and r is the distance between them.
- In our case, q1 = -2.73 μC and q2 = -9.56 μC. Converting μC to C:
q1 = -2.73 x 10^-6 C
q2 = -9.56 x 10^-6 C
- The distance (r) is given as 0.0465 m.
- Calculate the potential energy (PE) using the formula above.
2. Calculate the initial kinetic energy (KE) of the particle:
- The kinetic energy of the particle can be calculated using the formula:
KE = (1/2) * m * v^2
where m is the mass of the particle and v is its initial velocity.
- In our case, m = 9.50 x 10^-3 kg and v = 97.2 m/s.
- Calculate the initial kinetic energy (KE) using the formula above.
3. Apply the conservation of mechanical energy:
- At the closest approach, the kinetic energy of the particle is completely converted into the electrostatic potential energy. Therefore, at this moment, the sum of the initial kinetic energy and the potential energy is zero.
- Set the sum of the initial kinetic energy and the potential energy equal to zero and solve for the distance (r):
KE + PE = 0
4. Solve for the distance of closest approach (r).
Let's calculate step by step:
Step 1:
Using the formula for electric potential energy:
PE = k * (q1 * q2) / r
PE = (8.99 x 10^9 N m^2/C^2) * (-2.73 x 10^-6 C) * (-9.56 x 10^-6 C) / 0.0465 m
Step 2:
Using the formula for kinetic energy:
KE = (1/2) * m * v^2
KE = (1/2) * (9.50 x 10^-3 kg) * (97.2 m/s)^2
Step 3:
Apply the conservation of mechanical energy:
KE + PE = 0
KE + PE = 0
(1/2) * (9.50 x 10^-3 kg) * (97.2 m/s)^2 + PE = 0
Solve for PE:
PE = - (1/2) * (9.50 x 10^-3 kg) * (97.2 m/s)^2
Step 4:
Now plug in the values to find the distance of closest approach (r):
PE = (8.99 x 10^9 N m^2/C^2) * (-2.73 x 10^-6 C) * (-9.56 x 10^-6 C) / r
Set PE equal to the value calculated in step 3:
(8.99 x 10^9 N m^2/C^2) * (-2.73 x 10^-6 C) * (-9.56 x 10^-6 C) / r = - (1/2) * (9.50 x 10^-3 kg) * (97.2 m/s)^2
Now solve for r.
To determine the distance of closest approach between the moving particle and the fixed charge, we can use the concept of the conservation of mechanical energy.
First, let's calculate the initial mechanical energy of the moving particle, which is the sum of its kinetic energy and its electric potential energy:
Initial Mechanical Energy (E₁) = Initial Kinetic Energy (KE) + Initial Electric Potential Energy (PE)
The initial kinetic energy can be calculated using the formula:
KE = (1/2) * mass * velocity²
Substituting the given values:
KE = (1/2) * (9.50 x 10^(-3) kg) * (97.2 m/s)²
= 4.60 J (Joules)
The initial electric potential energy can be calculated using the formula:
PE = (k * |q₁ * q₂|) / r
where k is the electrostatic constant (k = 9 x 10^9 Nm²/C²), q₁ and q₂ are the charges of the particles, and r is the initial separation distance between the charges.
Substituting the given values:
PE = (9 x 10^9 Nm²/C²) * |(-2.73 x 10^(-6) C) * (-9.56 x 10^(-6) C)| / (0.0465 m)
= 448 J (Joules)
Note: The absolute value | | is used in the equation because the charges have opposite signs.
Now, we can find the initial mechanical energy:
E₁ = KE + PE
= 4.60 J + 448 J
= 452 J (Joules)
During the motion, the mechanical energy remains constant since there are no external forces involved. At the closest approach, all the mechanical energy is in the form of electric potential energy. Therefore, we can equate the initial mechanical energy to the electric potential energy at the closest approach (E₂):
E₁ = E₂
E₂ = (k * |q₁ * q₂|) / r'
where r' is the closest approach distance. Substituting the values:
452 J = (9 x 10^9 Nm²/C²) * |(-2.73 x 10^(-6) C) * (-9.56 x 10^(-6) C)| / r'
Now, solve for r':
r' = (9 x 10^9 Nm²/C²) * |(-2.73 x 10^(-6) C) * (-9.56 x 10^(-6) C)| / 452 J
Calculate the value of r' to find the distance of closest approach.
KE=PE
mv²/2=k•q1•q2/r
r=2•k•q1•q2/ mv² =
=2•9•10⁹•2.73•10⁻⁶•9.56•10⁻⁶/9.5•10⁻³•97.2²=
=5.23•10⁻³ m