u.r.m. in the collision of two bodies

Data

• Speed of the first body: V₁ = 72 km/h = 20 m/s
• Difference in time of departure between the two bodies ∆t = 15 s
• Instant in which the bodies meet: t = 90 s

Preliminary considerations

• Two bodies begin their motion at different times. The time difference between them is 15 s. Therefore, when the first body has been in motion for 90 s, the second one has only been 90 - 15 = 75 s
• For them to meet after 90 s, the distance traveled by the second body in 75 s must be the same as the distance the first one travels in 90 s
• Keep in mind that we can use the sign convention in rectilinear motion that allows us to use scalar magnitudes instead of vectors to describe the motion

Resolution

The expression that allows us to determine the position of each body as a function of the speed and of the time is:

If t₁ is the time that the first body is in motion, then for the first body we get:

$$x_1=x_{01}+v_1\cdot t_1$$ 

If t₂ is the time that the second body is in motion, then for the second body we get:

$$x_2=x_{02}+v_2\cdot t_2$$ 

Both bodies depart from the same point, therefore x₀₁ = x₀₂ = 0 , but they do so at different times: t₂  = t₁ - 15 s. . When they meet, they are at t₁ = 90 s and t₂ = 90 - 15 = 75 s. Additionally, when they are in the same position x₁= x₂, resulting in:

Ambos cuerpos parten del mismo punto, por tanto x₀₁ = x₀₂ = 0 , pero lo hacen en instante de tiempos distintos: t₂  = t₁ - 15 s. En el momento en que se encuentran t₁ = 90 s y t₂ = 90 - 15 = 75 s. Además, cuando se encuentran están en la misma posición, es decir, x₁ = x₂ , quedando:

$$v_1·t_1=v_2·t_2\Rightarrow v_2=\frac{v_1·t_1}{t_1-15}=\frac{20·90}{75}=\boxed{24 m/s}$$