A bird flies in the xy-plane with a position vector given by r→=(αt−βt3)i^+γt2j^, with α=2.4 ms,β=1.6 ms3 and γ=4.0 ms2. The positive y-direction is vertically upward. At the bird is at the origin.Calculate the velocity vector of the bird as a function of time.Calculate the acceleration vector of the bird as a function oftime.What is the bird's altitude(y-coordinate) as it flies over x=0 for the first time after ?

Question

A bird flies in the xy-plane with a position vector given by r→=(αt−βt3)i^+γt2j^, with α=2.4 ms,β=1.6 ms3 and γ=4.0 ms2. The positive y-direction is vertically upward. At  the bird is at the origin.Calculate the velocity vector of the bird as a function of time.Calculate the acceleration vector of the bird as a function oftime.What is the bird&#039;s altitude(y-coordinate) as it flies over x=0 for the first time after ?

Margot Mill · Accepted Answer

A bird flies in the xy-plane with a position vector given by r→=(αt−βt3)i^+γt2j^, with α=2.4 ms,β=1.6 ms3 and γ=4.0 ms2.  r→=(2.4t−1.6t3)i→+4.0t2j→  The positive y-direction is vertically upward. At t=0 the bird is at the origin. The velocity vector of the bird as a function of time(v=dr/dt).  v→=(2.4t−4.8t2)i→+8.0tj→  The acceleration vector of the bird as a function of time (a=dv/dt)  a→=(−9.6t)i→+8.0j→

Jeffrey Jordon · Accepted Answer

The velocity vector is  v(t)=(α−βt2)i^+(γt)j^  1. The position vector can be found by integrating the velocity vector.  r(t)=∫v(t)dt=(αt−βt33+C1)i^+(γt22+C2)j^  At t = 0, the bird is at the origin, so the integration constants can be determined.  r(0)=C1i^+C2y^=0  Therefore, C1 and C2 are equal to zero.  r(t)=(αt−βt33)i^+(γt22)j^  2. The acceleration vector is the derivative of the velocity vector with respect to time.  a(t)=dv(t)dt=−2βti^+γj^  3. We will first find the time when the x-component of the position vector is equal to zero.  αt−βt33=0  α=βt33  t=3αβ=2.12s  We will plug in this value into the y-component of the position vector.  y(t)=γt22  y(2.12)=4(2.12)22=9m

A bird flies in the xy-plane with a position vector given by \vec{r}=(\alpha t-\beta t^3)\hat{i}+\gamma t^2\hat{j}, with \alpha=2.4\ m/s,\beta=1.6\ m/

Answered question

Answer & Explanation

New Questions in Algebra II