Question

A rocket sled having an initial speed of 150 mi/hr is slowed by a channel of water. Assume that, during the breaking process, the acceleration is given by a(t)=-kv^2, where v is the velocity and k is a positive constant.
a. Find the velocity and position functions at time t.
b. if it requires a distance of 2000 ft to slow the sled to 15 mi/hr, determine the value of k.
c. Find how many seconds of braking are required to slow the sled to 15 mi/hr.

Answer 1

Answer:

Step-by-step explanation:

a) since acceleration can be expressed as a(t) = dv/dt

We use the chosen rule to obtain

dv/dt = dv/dx * dx/dt = v* dv/dx

We can now set v dv/dx = -kv^2

Taking v from both sides

dv/dx = -kv

Dv/v = -kdx

In (v)= kx+ c

V= e^kx+c

From v (o) = 150

V = 150e^kx

When distance x= 2000ft convert to miles gives 0.378788 mi then velocity v = 15 mi/he

15= 150. e ^ 0.378788k

Lm (o.t) = -0.378788k

K= 6.0788mi

C) we return to dv/dt = -kv^2 = dv/v^2 = -k dt

-1/v = -kt +c

From v(o) = 150, we get - 1/150= c and

1/v - 1/150= kt

We know v= 15mi/hr and k = 6.0788

1/15 - 1/150 = 6.0788t

T = 9.8704.10^-3hr.