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. Identify the units of the coefficient A, 2. physical meaning of A, 3. velocity of the car at t=0 and 4. asymptote of this function as t ⮕ ∞?

Answer 1

$v(t)=A(1-e^{-t/t_o})$ is the equation given for velocity, where;

$t_o$ = time to reach the maximum speed.

1.Except A all other terms in the right hand side of the equation are dimensionless constants.

Hence, A must be of the dimensions same as that of velocity.

$\implies$ A has unit same as velocity which is $(m/s)$ .

2.$lim_{v \to \infty} v(t)=A(1-e^{-\infty})=A(1-0)=A$ Thus physical significance of A is that it is the terminal velocity(as t tends to infinity, velocity tends to A)

3. $v(t)=A(1-e^{-t/t_o})$

$\implies v(0)=A(1-1)=0m/s$

4. As t tends to infinity,v(t) is same as terminal velocity that is equal to A.

Thus, asymptote of this function as $t \to ∞$ is $v=A$ .