9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.

d. A piecewise function for s(t)

Variable gravity At Earth’s surface, the acceleration due to gravity is approximately g=9.8 m/s² (with local variations). However, the acceleration decreases with distance from the surface according to Newton’s law of gravitation. At a distance of y meters from Earth’s surface, the acceleration is given by a(y) = - g / (1+y/R)², where R=6.4×10⁶ m is the radius of Earth.

b. Use the Chain Rule to show that dv/dt = 1/2 d/dy(v²).

9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.

c. The position at t=5

Position, displacement, and distance A projectile is launched vertically from the ground at t=0, and its velocity in flight (in m/s) is given by v(t)=20−10t. Find the position, displacement, and distance traveled after t seconds, for 0≤t≤4.

9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.

b. The distance traveled between t=0 and t=5

Piecewise velocity The velocity of a (fast) automobile on a straight highway is given by the function

$v(t)= \begin{cases}3 t & \text { if } 0 \leq t<20 \\ 60 & \text { if } 20 \leq t<45 \\ 240-4 t & \text { if } t \geq 45\end{cases}$

where is measured in seconds and v has units of m/s. 

d. What is the position of the automobile when t=75?

Fuel consumption A small plane in flight consumes fuel at a rate (in gal/min) given by

R'(t) ={ 4t^{1/3} if 0 ≤ t ≤ 8 (take-off)

2 if t> 0 (cruising)

a. Find a function R that gives the total fuel consumed, for 0≤t≤8.

2–3. Displacement, distance, and position Consider an object moving along a line with the following velocities and initial positions. Assume time t is measured in seconds and velocities have units of m/s.

d. Determine the position function s(t) using the Fundamental Theorem of Calculus (Theorem 6.1). Check your answer by finding the position function using the antiderivative method.

v(t) = 12t²-30t+12, for 0 ≤ t ≤ 3; s(0)=1

Flow rates in the Spokane River The daily discharge of the Spokane River as it flows through Spokane, Washington, in April and June is modeled by the functions

r1(t) = 0.25t²+37.46t+722.47 (April) and

r2(t) = 0.90t²−69.06t+2053.12 (June), where the discharge is measured in millions of cubic feet per day, and t=0 corresponds to the beginning of the first day of the month (see figure).

a. Determine the total amount of water that flows through Spokane in April (30 days).