Position vector of a point mass

The position vector of a point mass moving in a plane can be expressed in the established reference frame by the relation:
r(t) = (t²+ 2t + 1 ; 2t + 1),
where t is time in seconds and the vector coordinates are in meters. Calculate:
a) What is the position of the point mass at time t = 2 s?
b) The magnitude of the velocity of the point mass at time t = 4 s
c) The magnitude of the acceleration of the point mass at time t = 5 s

Final Answer:

x =  9 m
y =  5 m
v =  10.198 m/s
a =  2 m/s²

Step-by-step explanation:

$$\begin{gathered} t_1=2\text{s} \\ r(t)=(t_2+2t+1;2t+1)) \\ x=t_1^2+2\cdot t_1+1=2^2+2\cdot 2+1=9\text{m} \end{gathered}$$

$y=2\cdot t_1+1=2\cdot 2+1=5\text{m}$

$$\begin{gathered} t_2=4\text{s} \\ v(t)=r^{\prime }(t)=(2t+2;2) \\ {v}_0=2\cdot t_2+2=2\cdot 4+2=10 \\ {v}_1=2 \\ v=\sqrt{v_0^2+v_1^2}=\sqrt{10^2+2^2}=2\sqrt{26}=10.198\text{m/s} \end{gathered}$$

$$\begin{gathered} t_3=5\text{s} \\ a(t)=r^{\prime \prime }(t)=(2;0) \\ a=\sqrt{2^2+0^2}=2{\text{m/s}}^2 \end{gathered}$$

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