Particle with Repulsive Spring-like Force · USAPhO 2012 #B1
Physics · mechanics
Problem
A particle of mass m moves under a force similar to that of an ideal spring, except that the force repels the particle from the origin: F = +mα²x In simple harmonic motion, the position of the particle as a function of time can be written x(t) = A cos ωt+B sin ωt Likewise, in the present case we have x(t) = A f₁(t)+B f₂(t) for some appropriate functions f₁ and f₂. a. f₁(t) and f₂(t) can be chosen to have the form eʳᵗ. What are the two appropriate values of r? b. Suppose that the particle begins at position x(0) = x₀ and with velocity v(0) = 0. What is x(t)? c. A second, identical particle begins at position x(0) = 0 with velocity v(0) = v₀. The second particle becomes closer and closer to the first particle as time goes on. What is v₀?Intuitive explanation
This problem asks you to solve the equation of motion for a repulsive force instead of the usual restoring force. The key insight is that the math works almost identically to SHM—you still substitute an exponential ansatz into the differential equation—but now the characteristic equation yields real exponents instead of imaginary ones, giving exponential growth and decay rather than oscillation. Once you find those two exponents, you can build the general solution and use initial conditions to pin down the constants, then think about what it means for two particles to approach each other when one is being pushed away from the origin.
Intuitive explanation
This problem asks you to solve the equation of motion for a repulsive force instead of the usual restoring force. The key insight is that the math works almost identically to SHM—you still substitute an exponential ansatz into the differential equation—but now the characteristic equation yields real exponents instead of imaginary ones, giving exponential growth and decay rather than oscillation. Once you find those two exponents, you can build the general solution and use initial conditions to pin down the constants, then think about what it means for two particles to approach each other when one is being pushed away from the origin.