1. Calculus and Newton’s 2 nd Law Sometimes, we want to find a formula for the velocity (or position) of an object as a function of time. 公式

2. (1) Solve Newton’s 2 nd Law for the acceleration.

3. (2) Find the velocity by integrating the expression for the acceleration. 求它的积分

4. (2) Find the velocity by integrating the expression for the acceleration.

6. (3) If necessary, integrate again to find the position.

7. Example What is the least speed for a projectile launched vertically upward from the Earth’s surface, such that it will continue into space and never return to Earth? Assume that gravity is the only force felt after launch.

8. GIVEN: Mass of the object m Mass of the Earth m E Height of the object from the center of the Earth y(t) Gravitational force F g = Gm E m/y 2 (down) FIND: Initial speed v i such that the object never returns to the Earth.

9. PROCEDURE: (1) Use Newton’s 2 nd Law to find the acceleration of the object. (2) Integrate the acceleration to get an expression for the speed as a function of time. (3) Analyze this expression to find the minimum launch speed such that the velocity never changes direction.

10. Example When a small sphere, starting from rest, falls through a liquid, it experiences a drag force F D in addition to the force of gravity. F D is directed upwards and has a magnitude F D = bv, where b is a constant. Find the speed of the sphere as a function of time, v(t).

11. TRANSLATION: An object of known mass accelerates from rest, subject to known forces; find an expression for its speed as a function of time.

12. GIVEN: Mass of the object m Weight force F g = mg (down) Drag force F d = bv (up) FIND: Velocity v(t)

13. PROCEDURE: (1) Use Newton’s 2 nd Law to find the acceleration of the object. (2) Integrate the acceleration to get an expression for the velocity as a function of time.

14. It is nice to solve Newton’s 2 nd Law with calculus… but usually it doesn’t work! Example: The motion of three planets, interacting via gravity, has no analytical solution.

15. In such cases, we must solve Newton’s 2 nd Law numerically, using computers.

16. Newton’s 2 nd Law (update form): (Assumes that Δt is small enough for F to be almost constant.)

17. Example: compute the Earth’s motion around the Sun.

18. Step 1:

19. Step 2:

20. Step 3:

21. And so on…