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. Questions: 1. Create a graph of velocity vs time for this model? 2. Derive an equation x(t) for the instantaneous position of the car as a function of time. Identify the value of x when t= 0s and then asymptote of this function as t ⮕ ∞?

Answer 1

1. A graph of velocity vs time for this model.

2. An equation x(t) for the instantaneous position of the car as a function of time.

Position is given by equation:

$x(t) = \int v(t) dt = \int A(1-e^{-\frac{t}{t_{max}}})dt = x_0+At+At_{max}e^{-\frac{t}{t_{max}}}\\$

When t = 0: $x = x0 + At_{max}\\$

When $t \rightarrow \infin: x \rightarrow \infin\\$