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A proton follows the path shown in (Figure 1) . Its initial speed is v0 = 1.6×106 m/s .?

What is the proton's speed as it passes through point P?

I've gotten 2.56E6 and 2.0E6 as answers and mastering physics will not accept either one as correct. Please help!

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For neatness and convenience, I'll use the notation 1.6E-19 (for example) instead of 1.6x10⁻¹⁹.

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Proton's initial kinetic energy K1 = ½mv₀²

= ½ x 1.67E-27 x (1.6E6)²

= 2.138E-15J

The potential energy at the start point P1 = kqQ/r

= 9E9 x 1.6E-19 x -10E-9/0.003

= -4.8E-15 J

The potential energy at the end point P2 = kqQ/r

= 9E9 x 1.6E-19 x -10E-9/0.004

= -3.6E-15 J

Since energy is conserved:

K1 P1 = K2 P2

K2 = K1 P1 - P2

. . .= 2.138E-15 (-4.8E-15) - (-3.6E-15)

. . .= 9.38E-16 J

½mv² = 9.38E-16

v = √[2 x 9.38E-16 / 1.67E-27]

. .= 1.06E6 m/s

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Of course the solution can be written much quicker than that. I would usually just write:

½mv₀² kqQ/r1 = ½mv² kqQ/r2

½mv² = ½mv₀² kqQ(1/r1 - 1/r2)

and then substitute values.

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