AP Physics C: Mechanics:: Newtons Laws

How is NEWTON'S SECOND LAW expressed as a DIFFERENTIAL EQUATION, using the calculus definitions of velocity and acceleration, in terms of the NET FORCE and the object's MOMENTUM?

- **A)** Newton's second law states that F_net = dp/dx (net force equals momentum's derivative with respect to POSITION), rather than with respect to time
- **B)** Newton's second law is F_net = dp/dt (net force equals the TIME DERIVATIVE of momentum) -- for CONSTANT mass, this reduces to the more familiar F_net=ma, but the dp/dt form is more GENERAL, correctly handling systems where mass itself CHANGES over time
- **C)** Newton's second law applies only to constant-mass systems, with the dp/dt formulation being an entirely unrelated, invalid concept
- **D)** This concept has no actual mathematical relationship between Newton's second law and the derivative of momentum

For a VARIABLE FORCE F(x) that depends on POSITION (rather than being constant), how must NEWTON'S SECOND LAW be solved using calculus to find an object's VELOCITY as a function of position, given the differential equation m*(dv/dt) = F(x)?

- **A)** This concept has no actual relationship between the chain rule and solving Newton's second law for a position-dependent force
- **B)** Using the CHAIN RULE, dv/dt can be rewritten as v*(dv/dx) (since dv/dt = dv/dx * dx/dt = dv/dx * v) -- this transforms the equation into m*v*(dv/dx) = F(x), which can then be solved by SEPARATING VARIABLES and INTEGRATING with respect to position x
- **C)** Variable position-dependent forces cannot actually be analyzed using any calculus-based method, making Newton's second law inapplicable in this case
- **D)** The equation m*(dv/dt)=F(x) can be solved by simply integrating F(x) with respect to TIME directly, without any need for the chain-rule substitution

For a VARIABLE FORCE F(v) that depends on VELOCITY (such as a DRAG FORCE, common for objects moving through a fluid), how is NEWTON'S SECOND LAW, m*(dv/dt) = F(v), typically solved using calculus?

- **A)** The differential equation m*(dv/dt) = F(v) is solved by SEPARATING VARIABLES (rearranging to isolate all velocity terms on one side and time terms on the other: m*dv/F(v) = dt), then INTEGRATING both sides to find velocity as a function of time
- **B)** Velocity-dependent forces cannot actually be analyzed using any calculus-based method, making Newton's second law inapplicable in this case
- **C)** This concept has no actual relationship between separation of variables and solving Newton's second law for a velocity-dependent force
- **D)** The equation m*(dv/dt)=F(v) can only be solved if F(v) happens to be CONSTANT, with no actual method available for a genuinely velocity-dependent force

For an object falling through air with a LINEAR DRAG FORCE (F_drag = -b*v, proportional to velocity and opposing motion), what does the resulting DIFFERENTIAL EQUATION, m*(dv/dt) = mg - b*v, predict about the object's LONG-TERM (steady-state) behavior as time approaches infinity?

- **A)** As time increases, the object approaches a CONSTANT TERMINAL VELOCITY, v_terminal = mg/b, at which point the gravitational force is EXACTLY BALANCED by the drag force (dv/dt=0, since the net force becomes zero at this specific velocity)
- **B)** The object's velocity always returns to ZERO as time increases, regardless of the drag force or gravitational force involved
- **C)** The object's velocity increases WITHOUT BOUND (approaching infinity) as time increases, contradicting the actual terminal-velocity-limiting behavior of drag forces
- **D)** This differential equation has no actual relationship to predicting an object's long-term velocity behavior while falling through air

What is the mathematical FORMULA for FRICTION as a force that OPPOSES relative motion (or the TENDENCY of relative motion) between two surfaces, and how does this concept extend from the algebra-based treatment (familiar from AP Physics 1) into this course's calculus-based framework?

- **A)** Friction (F_friction = mu*N, where mu is the coefficient of friction and N is the normal force) retains the SAME basic formula from algebra-based physics, but in this course's calculus-based framework, friction is INCORPORATED directly into the differential equation of motion (F_net=ma) to solve for velocity and position as functions of time, especially when combined with other variable forces
- **B)** Friction has no actual relationship to this course's calculus-based differential-equation approach to solving for motion
- **C)** The friction formula F=mu*N is COMPLETELY DIFFERENT and unrelated in this calculus-based course compared to the algebra-based treatment in AP Physics 1
- **D)** Friction can only be analyzed using algebra-based methods, with no actual applicability of calculus techniques to friction-involving problems

For a system involving an object launched with an INITIAL velocity v0, subject to a LINEAR DRAG FORCE (F_drag=-b*v) and NO other forces (e.g., a puck sliding on a surface with air resistance, no friction), what is the resulting VELOCITY function v(t), found by SOLVING the differential equation m*(dv/dt) = -b*v via separation of variables?

- **A)** v(t) = v0 - (b/m)*t (a LINEAR decrease), incorrectly treating the differential equation as if it had constant (rather than velocity-dependent) deceleration
- **B)** This differential equation has no actual solvable closed-form velocity function using calculus methods
- **C)** v(t) = v0*(b/m)*t, incorrectly combining the initial velocity and the drag coefficient in a way that doesn't correctly solve the differential equation
- **D)** Separating variables: dv/v = -(b/m)*dt; integrating both sides: ln|v| = -(b/m)*t + C; solving for v and applying v(0)=v0 gives the EXPONENTIAL DECAY function, v(t) = v0*e^(-(b/m)*t)

What is a SPRING FORCE (following Hooke's Law, F=-kx), and how does INCORPORATING this VARIABLE, POSITION-DEPENDENT force into Newton's second law (m*(d^2x/dt^2) = -kx) produce a DIFFERENTIAL EQUATION whose solution describes SIMPLE HARMONIC MOTION?

- **A)** The spring force equation has no actual relationship to producing a differential equation whose solution describes simple harmonic motion
- **B)** Substituting F=-kx into Newton's second law gives m*(d^2x/dt^2) = -kx, a SECOND-ORDER DIFFERENTIAL EQUATION whose GENERAL SOLUTION is a sinusoidal function of the form x(t)=A*cos(omega*t+phi), where omega=sqrt(k/m) -- this differential equation approach provides a rigorous derivation of simple harmonic motion directly from Newton's second law
- **C)** This differential equation can only be solved using algebra-based methods, with no actual role for calculus in deriving the simple-harmonic-motion solution
- **D)** Substituting F=-kx into Newton's second law produces an equation with NO sinusoidal solutions, contradicting the well-established simple-harmonic-motion behavior of spring systems

What is a NORMAL FORCE, and how does its DIRECTION (always PERPENDICULAR to a contact surface) remain a KEY CONCEPTUAL TOOL for setting up FREE-BODY DIAGRAMS in this calculus-based course, just as it was in algebra-based physics?

- **A)** Normal force concepts from algebra-based physics have no actual relevance or applicability to this calculus-based course
- **B)** A normal force is the force a surface exerts PERPENDICULAR to itself on an object in contact with it; correctly identifying the normal force's direction remains ESSENTIAL for setting up free-body diagrams and the resulting differential equations of motion in this calculus-based course, exactly as it was in the earlier algebra-based treatment
- **C)** Free-body diagrams are ENTIRELY UNNECESSARY in this calculus-based course, with differential equations somehow being set up without needing to first identify all forces acting on an object
- **D)** A normal force always acts PARALLEL (rather than perpendicular) to a contact surface, contradicting its actual, well-established perpendicular direction