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).

{Use of Tech} Oscillating motion A mass hanging from a spring is set in motion, and its ensuing velocity is given by v(t) = 2π cos πt, for t≥0. Assume the positive direction is upward and s(0)=0. 

a. Determine the position function, for t≥0.

{Use of Tech} Oscillating motion A mass hanging from a spring is set in motion, and its ensuing velocity is given by v(t) = 2π cos πt, for t≥0. Assume the positive direction is upward and s(0)=0. 

c. At what times does the mass reach its low point the first three times? 

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. 

c. What is the distance traveled by the automobile in the first 60 s?

A running model A model for the startup of a runner in a short race results in the velocity function v(t) = a(1 - e⁻ᵗ/ᶜ), where a and c are positive constants, t is measured in seconds, and v has units of m/s. (Source: Joe Keller, A Theory of Competitive Running, Physics Today, 26, Sep 1973)

b. Using the velocity in part (a) and assuming s(0) = 0, find the position function s(t), for t ≥ 0.

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

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)