A proton is put on the positive Y axis a large distance (100km) from the origin. What is the minimum speed required in the - Y direction so the proton will just reach the point (0,20) cm?
Is there some sort of repelling positive charge at the origin or what ?
If so the kinetic energy at infinity has to equal the potential energy at 20 cm
0,0 is the origin there is no repelling charge
Then there is no force on the proton and it will keep going forever with infintesimal velocity.
So what's the minimum speed ?
To solve this problem, we can use the principles of physics, particularly the concept of conservation of energy. We'll need to calculate the gravitational potential energy and the kinetic energy of the proton at position (0, 20 cm) in order to determine the minimum speed required in the -Y direction.
First, let's determine the gravitational potential energy of the proton at point (0, 20 cm). The gravitational potential energy (U) is given by the formula:
U = - G * (m1 * m2) / r
Here, G is the gravitational constant, m1 is the mass of the proton, m2 is the mass of the Earth, and r is the distance between the proton and the origin.
Since we are assuming that the proton is very close to the origin, we can safely ignore the change in gravitational potential energy as it moves from a large distance to the origin. Therefore, we can write the expression for the gravitational potential energy at the point (0, 20 cm) as:
U = - G * (m1 * m2) / r
Next, let's calculate the kinetic energy of the proton at point (0, 20 cm). The kinetic energy (K) is given by the formula:
K = (1/2) * m1 * v^2
Here, m1 is the mass of the proton, and v is the velocity of the proton in the -Y direction.
Since we need to find the minimum speed required, we can equate the initial gravitational potential energy with the final kinetic energy:
U_initial = K_final
Plugging in the values, we have:
- G * (m1 * m2) / r_initial = (1/2) * m1 * v^2
Simplifying the equation, we get:
v^2 = -2 * G * m2 / r_initial
Taking the square root of both sides gives us the minimum speed required in the -Y direction:
v = sqrt(-2 * G * m2 / r_initial)
Now, let's plug in the values to get the final answer:
- The gravitational constant, G = 6.67430 * 10^(-11) N(m/kg)^2
- The mass of the proton, m1 = 1.6726219 * 10^(-27) kg
- The mass of the Earth, m2 = 5.97237 * 10^24 kg
- The distance between the proton and the origin, r_initial = 100,000,000 cm
Substituting the values, we can calculate the minimum speed required in the -Y direction.