Motion with Decreasing Acceleration

The acceleration of a mass point during its rectilinear movement decreases uniformly from the initial value a0 = 10 m/s2 at time t0 = 0 to a zero value for a period of 20 s. What is the speed of the mass point at time t1 = 20 s, and what is the path of the mass during that time? Did the point escape when it was at rest at time t0?

Final Answer:

v₂₀ =  190 m/s

Step-by-step explanation:

$$\begin{gathered} a_0=10{\text{m/s}}^2 \\ {t}_0=0\text{s} \\ {v}_0=0\text{m/s} \\ {a}_{20}=0 \\ {t}_{20}=20\text{s} \\ a=\frac{dv}{dt} \\ v=\frac{ds}{dt} \\ a(t)=q+w\cdot t \\ {a}_0=q+w\cdot t_0 \\ {a}_{20}=q+w\cdot t_{20} \\ 10=q+w\cdot 0 \\ 0=q+w\cdot 20 \\ q=10 \\ q+20w=0 \\ Row2-Row1\to Row2 \\ 20w=-10 \\ w=\frac{-10}{20}=-0.5 \\ w=\frac{-1}{2}=-0.5 \\ a(t)=10-1/20\cdot t{\text{m/s}}^2 \\ v(t)=integrala(t) \\ v(t)=10\cdot t-1/20\cdot t^2/2+c \\ v(t)=10\cdot t-t^2/40+c \\ {v}_{20}=10\cdot t_{20}-t_{20}^2/40+v_0=10\cdot 20-20^2/40+0=190\text{m/s} \end{gathered}$$

Try another example