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

Question

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

Emily Ammerman · Accepted Answer

# vec v = << -3e^(-2), 2e^2-4 >> #
# \ \ \ \ ~~ << -0.406, 10.778 >> #

We have the following displacement function: # f(t) = (e^(t-t^2), 2e^t-t^2) =(x(t), y(t))#, say Then differentiating wrt #t# we get: # dx/dt = (1-2t)e^(t-t^2) # # dy/dt = 2e^t-2t # And so when #t=2# we have: # dx/dt = (1-4)e^(2-4) = -3e^(-2) \ \ (~~-0.406) # # dy/dt = 2e^2-4 \ \ (~~10.778) # Hence, assuming a standard Cartesian coordinate system, the instantaneous velocity when #t=2# is the vector: # vec v = << -3e^(-2), 2e^2-4 >> # Note: If we wanted the instantaneous speed , then this would be given by: # v = || vec v || = sqrt( (-3e^(-2))^2 + (2e^2-4)^2 ) #

Christopher Allers · Answer

The instantaneous velocity is #=(-0.41,10.78)#

The velocity is the derivative of the position.

#f(t)=(e^(t-t^2), 2e^t-t^2))#

#v(t)=f'(t)=((1-2t)e^(t-t^2),2e^t-2t))#

Therefore,

#v(2)=(-3e^-2, 2e^2-4)#

#=(-0.41,10.78)#

Paisley Johnson · Answer

undefined To find the instantaneous velocity of an object moving according to the function $ f(t) = (e^{t - t^2}, 2e^t - t^2) $ at $ t = 2 $, we need to take the derivative of the function with respect to $ t $ and then evaluate it at $ t = 2 $.

First, find the derivative of each component of $ f(t) $ with respect to $ t $:

$ f(t) = (e^{t - t^2}, 2e^t - t^2) $

$ \frac{df_1}{dt} = \frac{d}{dt}(e^{t - t^2}) $

$ \frac{df_2}{dt} = \frac{d}{dt}(2e^t - t^2) $

Now, calculate the derivatives:

$ \frac{df_1}{dt} = e^{t - t^2} \cdot (1 - 2t) $

$ \frac{df_2}{dt} = 2e^t - 2t $

Evaluate the derivatives at $ t = 2 $:

$ \frac{df_1}{dt}\bigg|_{t=2} = e^{2 - 2^2} \cdot (1 - 2(2)) = e^{-2} \cdot (1 - 4) = -3e^{-2} $

$ \frac{df_2}{dt}\bigg|_{t=2} = 2e^2 - 2(2) = 2e^2 - 4 $

So, the instantaneous velocity of the object at $ t = 2 $ is $ \left(-3e^{-2}, 2e^2 - 4\right) $.