Unit 1: Kinematics with Calculus

To try this unit free of charge, click here.

Big Ideas:
1. The four basic parameters that describe motion are position, velocity, acceleration, and time.
2. Given initial conditions, the motion of a particle can be modeled mathematically. 
3. Vectors are measureable quantities that have both magnitude and direction.
4. The trajectory of a particle in a uniform gravitational field is parabolic.
5. Objects in uniform circular motion must be centripetally accelerated.

Learner Objectives (as given by the College Board):
1. Students should understand the general relationships among position, velocity, and acceleration for the motion of a particle along a straight line, so that:
A. Given a graph of one of the kinematic quantities, position, velocity, or acceleration, as a function of time, they can recognize in what time intervals the other two are positive, negative, or zero, and can identify or sketch a graph of each as a function of time.
B. Given an expression for one of the kinematic quantities, position, velocity, or acceleration, as a function of time, they can determine the other two as a function of time, and find when these quantities are zero or achieve their maximum and minimum values.

2. Students should know how to deal with situations in which acceleration is a specified function of velocity and time so they can write an appropriate differential equation and solve it for u by separation of variables, incorporating correctly a given initial value of u.

3. Students should understand the special case of motion with constant acceleration, so they can:
A. Write down expressions for velocity and position as functions of time, and identify or sketch graphs of these quantities.
B. Use the equations v = v_o + at, x = x_o + v_ot + ½at², and v² = v_o² + 2a(x - x_o) to solve problems involving one-dimensional motion with constant acceleration.

4. Students should understand the general motion of a particle in two dimensions so that, given functions x(t) and y(t) which describe this motion, they can determine the components, magnitude, and direction of the particle’s velocity and acceleration as functions of time.

5. Students should be able to add, subtract, and resolve displacement and velocity vectors, so they can:
A. Determine components of a vector along two specified, mutually perpendicular axes.
B. Determine the net displacement of a particle or the location of a particle relative to another.
C. Determine the change in velocity of a particle or the velocity of one particle relative to another.

6. Students should understand the motion of projectiles in a uniform gravitational field, so they can:
A. Write down expressions for the horizontal and vertical components of velocity and position as functions of time, and sketch or identify graphs of these components.
B. Use these expressions in analyzing the motion of a projectile that is projected with an arbitrary initial velocity.

7. Students should understand the uniform circular motion of a particle, so they can:
A. Relate the radius of the circle and the speed or rate of revolution of the particle to the magnitude of the centripetal acceleration
B. Describe the direction of the particle’s velocity and acceleration at any instant during the motion.
C. Determine the components of the velocity and acceleration vectors at any instant, and sketch or identify graphs of these quantities.