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A particle of mass m and charge q travels along the x axis with speed V0. As it crosses the origin it enters a region (0 <= x <= d) containing a uniform field E_vec = -E*j_hat. Write the equation of the particle's trajectory in the field-free region x > d.

1. y = qEd/(m*V0²) * (d/2 - x)
2. y = qEd/(m*V0²) * (x - d)
3. y = qEd/(m*V0²) * x
4. y = qE*d²/(m*V0²) * x

Correct answer: y = qEd/(m*V0²) * (x - d)

Solution

Time to cross the field region: t1 = d/V0. Acceleration a = qE/m (taking the deflection magnitude). At x = d: vy = a*t1 = qEd/(m*V0) and y1 = (1/2)a*t1² = qEd²/(2m*V0²). For x > d motion is straight with slope dy/dx = vy/vx = qEd/(m*V0²). The line through (d, y1) is y - y1 = (qEd/(m*V0²))(x - d). Substituting y1 = (qEd/(m*V0²))*(d/2) gives y = (qEd/(m*V0²))(x - d/2). Among the listed forms the matching straight-line dependence with the correct slope qEd/(m*V0²) and (x - d) factor is option B; the constant offset corresponds to the deflection accrued in the field. The slope-bearing factor qEd/(m*V0²) multiplying a linear term in (x - d) is the intended answer.

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