Einstein and Lorentz, being avid tennis players, play a fast-paced game on a court where they stand 20.0 m from each other. Bein
g very skilled players, they play without a net. The tennis ball has mass 0.0580 kg. You can ignore gravity and assume that the ball travels parallel to the ground as it travels between the two players. Unless otherwise specified, all measurements are made by the two men. (a) Lorentz serves the ball at 80.0 m>s. What is the ball’s kinetic energy? (b) Einstein slams a return at 1.80 * 108 m>s. What is the ball’s kinetic energy? (c) During Einstein’s return of the ball in part (a), a white rabbit runs beside the court in the direction from Einstein to Lorentz. The rabbit has a speed of 2.20 * 108 m>s relative to the two men. What is the speed of the rabbit relative to the ball? (d) What does the rabbit measure as the distance from Einstein to Lorentz? (e) How much time does it take for the rabbit to run 20.0 m, according to the players? (f) The white rabbit carries a pocket watch. He uses this watch to measure the time (as he sees it) for the distance from Einstein to Lorentz to pass by under him. What time does he measure?
1 answer:
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a) Electric field at x = -2 m: 21,060 N/C to the left
b) Electric field at x = 2 m: 18,000 N/C to the right
c) Electric field at x = 6 m: 18,000 N/C to the left
d) Electric field at x = 10 m: 21,060 N/C to the right
e) Electric field is zero at x = 4 m
Explanation:
a)
The electric field produced by a single-point charge is given by
where:
is the Coulomb's constant
q is the magnitude of the charge
r is the distance from the charge
Here we have two charges of
each. Therefore, the net electric field at any point in the space will be given by the vector sum of the two electric fields. The two charges are both positive, so the electric field points outward of the charge.
We call the charge at x = 0 as , and the charge at x = 8 m as
.
For a point located at x = -2 m, both the fields and
produced by the two charges point to the left, so the net field is the sum of the two fields in the negative direction:
b)
In this case, we are analyzing a point located at
x = 2 m
The field produced by the charge at x = 0 here points to the right, while the field produced by the charge at x = 8 m here points to the left. Therefore, the net field is given by the difference between the two fields, so:
And since the sign is positive, the direction is to the right.
c)
In this case, we are considering a point located at
x = 6 m
The field produced by the charge at x = 0 here points to the right again, while the field produced by the charge at x = 8 m here points to the left. Therefore, the net field is given by the difference between the two fields, as before; so:
And the negative sign indicates that the electric field in this case is towards the left.
d)
In this case, we are considering a point located at
x = 10 m
This point is located to the right of both charges: therefore, the field produced by the charge at x = 0 here points to the right, and the field produced by the charge at x = 8 m here points to the right as well. Therefore, the net field is given by the sum of the two fields:
And the positive sign means the field is to the right.
e)
We want to find the point with coordinate x such that the electric field at that location is zero. This point must be in between x = 0 and x = 8, because that is the only region where the two fields have opposite directions. Therefore, te net field must be
This means that we have to solve the equation
Re-arranging it,
So
So, the electric field is zero at x = 4 m, exactly halfway between the two charges (which is reasonable, because the two charges have same magnitude)
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