What is the speed of the waves on the rope (in m/s)?

Question

Two children stretch a jump rope between them and send wave pulses back and forth on it. The rope is 2.7 m long, its mass is 0.46 kg, and the force exerted on it by the children is 43 N.

Answer 1

You need to know speed of the transversal wave (this is the type of wave children generate, perhaps unknowingly)

v = √(T / ρ)

where T is force of tension in the rope in [ N ], and ρ is linear mass density in [ kg / m ]

This formula is not too trivial to derive. It comes out from the differential equation describing propagation of waves in media. However, we still can make sense of it. You know from experience that the tighter the rope, the faster is the wave. Elasticity is what makes a wave. If children slack off, there will be no wave at all. You also can accept that the heavier mass moves slowly due to its inertia property. So if you forget the exact formula, you still might remember that wave speed is proportional to T / ρ, not ρ / T, and not T * ρ.

OK, anyways:

v = √(T / ρ) = √(T * L / m) = √(43N * 2.7m / 0.46kg) = 15.9m/s