Question

Using the mathematical model: to calculate velocity of a car accelerating from rest in straight line. Equation is v(t) = A (1-e ^ -t/t max speed) the ^ means to the power of. v(t) is the instantaneous velocity of the car (m/s), t is the time in seconds, tmaxspeed is the time to reach the maximum speed in seconds and A is a constant. Find out how to 1. Create a graph of position vs time for the given model? 2. Derive an equation a(t) for the instantaneous acceleration of the car as a function of time. Identify the acceleration of the car at t=0s? and asymptote of this function as t ⮕ ∞?

Answer 1

We are given velocity as a function of time so integrating it with respect to time we will get he equation of displacement with respect to time

$v(t)=A(1-e^{-t/t_{max}})$

$\intop{dx}=\intop A(1-e^{-t/t_{max}})dt$

$x(t)=A$ $t$ $+A(e^{-t/t{max}})t_{max}$

graph of the above function will be a linear one as e^-t/tmax will be almost equal to zero

Differentiating velocity with respect to time we will get the acceleration

$a(t)=d$ $(A(1-e^{-t/t_{max}}))/dt$

$a(t)=Ae^{-t/t_{max}}/t_{max}$

at t=0

acceleration= A/t_max

and as we tend towards infinity we see that the value of acceleration tends to become closer to zero hence the asymptote of this function is x-axis or a=0