Answer to Question #94823 in Mechanics | Relativity for stefanie

Question #94823
Suppose the position vector for a particle is given as a function of time by
r
(t) = x(t)î + y(t)ĵ, with x(t) = at + b and y(t) = ct2 + d, where a = 2.00 m/s, b = 1.40 m, c = 0.129 m/s2, and d = 1.12 m.
(a) Calculate the average velocity during the time interval from t = 2.20 s to t = 4.05 s.
v
= m/s

(b) Determine the velocity at t = 2.20 s.
v
= m/s

Determine the speed at t = 2.20 s.
m/s
1
Expert's answer
2019-09-19T09:49:49-0400

The position vector for a particle is given by equation


r(t)=x(t)i^+y(t)j^{\bf r}(t)=x(t){\bf\hat i}+y(t){\bf\hat j}

r(t)=(at+b)i^+(ct2+d)j^{\bf r}(t)=(at+b){\bf\hat i}+(ct^2+d){\bf\hat j}

r(t)=(2.00t+1.40)i^+(0.129t2+1.12)j^{\bf r}(t)=(2.00t+1.40){\bf\hat i}+(0.129t^2+1.12){\bf\hat j}

(a) The average velocity of a particle

vave=r2r1t2t1{\bf v}_{ave}=\frac{{\bf r_2}-{\bf r_1}}{t_2-t_1}

r1=r(t1)=(2.00×2.20+1.40)i^+(0.129×(2.20)2+1.12)j^{\bf r}_1={\bf r}(t_1)=(2.00\times 2.20+1.40){\bf\hat i}+(0.129\times (2.20)^2+1.12){\bf\hat j}

=5.80i^+1.74j^=5.80{\bf\hat i}+1.74{\bf\hat j}

r2=r(t2)=(2.00×4.05+1.40)i^+(0.129×(4.05)2+1.12)j^{\bf r}_2={\bf r}(t_2)=(2.00\times 4.05+1.40){\bf\hat i}+(0.129\times (4.05)^2+1.12){\bf\hat j}

=9.50i^+3.24j^=9.50{\bf\hat i}+3.24{\bf\hat j}

So


vave=(9.50i^+3.24j^)(5.80i^+1.74j^)4.052.20=2i^+0.811j^{\bf v}_{ave}=\frac{(9.50{\bf\hat i}+3.24{\bf\hat j})-(5.80{\bf\hat i}+1.74{\bf\hat j})}{4.05-2.20}=2{\bf\hat i}+0.811{\bf\hat j}

(b) The velocity of a particle

v(t)=r(t)=[(2.00t+1.40)i^+(0.129t2+1.12)j^]{\bf v}(t)={\bf r}'(t)=[(2.00t+1.40){\bf\hat i}+(0.129t^2+1.12){\bf\hat j}]'

=2.00i^+0.258tj^=2.00{\bf\hat i}+0.258t{\bf\hat j}

v(2.20)=2.00i^+0.258×2.20j^=2.00i^+0.568j^{\bf v}(2.20)=2.00{\bf\hat i}+0.258\times 2.20{\bf\hat j}=2.00{\bf\hat i}+0.568{\bf\hat j}

The speed of a particle


v(2.20)=2.002+0.5682=2.08m/s|{\bf v}(2.20)|=\sqrt{2.00^2+0.568^2}=2.08\:\rm m/s


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