After the launch, the cart’s speed is v2. Beginning at time t = 0, the cart experiences a braking force of F = -bv, rest in comments

Question

where b is a positive constant with units of kg/s and v is the speed of the cart. Express your answers to the following in terms of m, b, v2, and physical constants, as appropriate. (5) a. Using Newton’s second law, write but DO NOT solve a differential equation that represents the motion of the cart while it experiences the braking force. b. Show that the speed v(t) of the cart as a function of time is given by the equation v(t) = v2 e ^-bt/m. c. Derive an expression for the distance the car travels from t = 0 until the time it comes to a stop.

Answer 1

For the function

f(t) = v₂ e^-bt/m

it will be useful below to first compute

the derivative f'(t) = -v₂b/m e^-bt/m

the indefinite integral F(t) = ∫f(t)dt = -v₂m/b e^-bt/m

a)

F = ma = m dv/dt

-bv = m dv/dt

b)

That is shown by

1) plugging into the above equation both v and dv/dt (which was computed at the top).

2) Observing that v(0) = v₂, which satisfies the boundary condition

c)

The cart comes to a stop at time t satisfying v(t) = 0, which never happens.

That means that the cart never stops.

However, in that infinite time it travels only a finite distance given by

the definite integral from 0 to ∞ of v(t).

That is, the distance

s = F(∞) - F(0) = v₂ m/b