(a) Given
\(\vec r=16t\hat i+25t^2\hat j+33\hat k\) cm
\(\vec r=0.16t\hat i+.25t^2\hat j+.33\hat k\) m
Velocity v = \(\frac{d\vec r}{dt}\)
= \(\frac{d}{dt}(0.16t\hat i+.25t^2\hat j+.33\hat k)\)
\(\vec v=0.16\hat i+0.50t\hat j+0\)
\(\vec v=0.16\hat i+0.50t\hat j\)
\(\hat v=0.16\hat i+0.50t\hat j\) m/s
acceleration
a = \(\frac{dv}{dt}\)
a = \(\frac{d}{dt}(0.16\hat i+0.50t\hat j)\)
a = 0 + 0.50 j
a = 0.50j m/s2
(b) \(\vec r=10sin15t\hat i+35t\hat j+e^{6t}\hat k\)
Velocity v = \(\frac{d\vec r}{dt}\)⇒ \(\frac{d}{dt}(10sin15t\hat i+35t\hat j+e^{6t}\hat k)\)
\(\vec V=10cos15t\times15\hat i+35\hat j+e^{6t}.6\hat k\)
\([\vec v=150cos15t\hat i+35\hat j+6e^{6t}\hat k]\) cm/sec
acceleration a = \(\frac{d\vec v}{dt}\)
a = \(\frac{d}{dt}(15 cos15t\hat i+35\hat j+6e^{6t}\hat k)\)
a = 150(-sin15t)15\(\hat i\) + 0 + 36e6t\(\hat k\)
[a = -2250 sin 15t \(\hat i\) + 36 e6k\(\hat k\)]