# What is the instantaneous velocity of an object moving in accordance to # f(t)= (t-e^t,e^(2t)) # at # t=2 #?

##### 1 Answer

Feb 22, 2017

Instantaneous velocity is given by the vector:

# vec(v) = << 1-e^2, 2e^4 >> # , or,#(1-e^2) hat(i)+2e^4hat(j)# , or,#( (1-e^2),(2e^4) )#

#### Explanation:

We have:

# f(t) = ( x(t), y(t) )# where#x(t)=t-e^t# ,#y(t)=e^(2t)#

Where

# dx/dt = 1-e^t = 1/(2sqrt(t+2))#

# dy/dt = 2e^(2t) #

So, when

# dx/dt = 1-e^2 ~~ -6.389#

# dy/dt = 2e^4 \ \ \ \ \ ~~ 109.196#

And so the instantaneous **velocity** is given by the vector:

# vec(v) = << 1-e^2, 2e^4 >> # , or,#(1-e^2) hat(i)+2e^4hat(j)# , or,#( (1-e^2),(2e^4) )#

If we want the instantaneous **speed** , it is given by:

# v = || vec(v) || #

# \ \ \= sqrt( (1-e^2)^2 + (2e^4)^2 ) #

# \ \ \= sqrt( 11964.651 ... ) #

# \ \ \~~ 109.383 ... #