How do gravity, circular-orbit speed, period and escape speed relate outside a spherically symmetric central body?

For a test mass m ≪ M outside an isolated spherically symmetric mass M, F = GMm/r² and g = GM/r². Circular orbits have v_orb = √(GM/r) and T² = 4π²r³/(GM). Escape from radius R under gravity alone requires v_esc = √(2GM/R), equal to √2 times the circular speed at that same radius. Ignore atmospheric drag and source rotation.

What are the apex properties, flight time, height and range of an ideal level-ground projectile?

With uniform downward g, no air resistance, launch speed v0 and angle 0 < θ < 90°, the apex has vx = v0 cosθ, vy = 0, ax = 0 and ay = −g. Returning to launch height takes T = 2v0 sinθ/g. Height above launch is H = v0² sin²θ/(2g), and range is R = v0² sin(2θ)/g. At fixed v0, range is greatest at 45°; complementary angles give equal ranges.

Where is a falling block’s speed greatest while compressing an ideal vertical spring?

For mass m dropped from rest a height h ≥ 0 above an unstretched massless spring of stiffness k, neglect drag and dissipation. At downward compression y, a = g − ky/m, so speed peaks at y = mg/k. From mg(h + y) = ½mv² + ½ky², v_max = √(2gh + mg²/k). Maximum speed occurs after first contact, while the spring remains in its linear range.

How is momentum conservation resolved for a two-dimensional glancing collision?

Suppose m1 initially moves along +x at speed v1i, m2 is stationary, and external impulse is negligible. With final speeds v1f and v2f at angles θ1 above and θ2 below +x: m1v1i = m1v1f cosθ1 + m2v2f cosθ2, and 0 = m1v1f sinθ1 − m2v2f sinθ2. These momentum equations do not require an elastic collision.

What is centripetal acceleration, and what speed keeps an unrestrained particle in contact at the top of an inside loop?

The inward radial acceleration is ac = v²/r = ω²r; “centripetal force” names the net inward force, not an extra interaction. At the top of a circular inside track of radius R, N + mg = mv²/R. Requiring N ≥ 0 gives v_min = √(gR). This contact condition assumes gravity and the track normal are the only radial forces.

What do Newton’s three laws say in an inertial frame, and what is a third-law pair?

Zero net force gives constant velocity. For a particle of constant mass, ΣF = dp/dt = ma, with vector quantities. For an interaction between A and B, F_AonB = −F_BonA. These equal and opposite forces act on different bodies. Weight and normal force on the same book are not a third-law pair.

How can linearizing a physics relation reveal a constant from a fitted slope?

For a small-angle simple pendulum, T² = (4π²/g)L. Plot T² against L: the ideal slope S gives g = 4π²/S. Estimate S from a fit; if using a graph, use well-separated points on the fitted line. The ideal intercept is zero, but do not force measured data through zero without justification. For constant acceleration from rest, plotting v² against Δx gives slope 2a.

What are average and instantaneous mechanical power?

Average power for work W over time Δt is P_avg = W/Δt. Instantaneous power is P = dW/dt. For a force acting on a particle with velocity v, P = F·v = Fv cosθ. Power measures the rate of energy transfer by work; 1 W = 1 J/s.

What forces and acceleration apply to a block sliding down a fixed rough incline?

For slope angle θ above horizontal, gravity has components mg sinθ down the slope and mg cosθ into it. With no other applied force and no normal acceleration, N = mg cosθ. In the kinetic-friction model, fk = μkN acts uphill. Taking downhill as positive, a = g(sinθ − μk cosθ) while the block slides downhill; a may be negative.

How do momentum conservation and collision types relate?

Total momentum is constant when net external force is zero; negligible external impulse suffices across a short collision. Elastic collisions conserve kinetic energy. Dissipative inelastic collisions reduce it; superelastic collisions increase it by releasing internal energy. In the point-mass model, perfectly inelastic objects share a final velocity, minimizing final translational kinetic energy for the given total momentum and masses.

How do position, velocity and acceleration relate through calculus and motion graphs?

In one dimension, v = dx/dt and a = dv/dt. Signed areas give Δx = ∫v dt and Δv = ∫a dt over the time interval. Where v ≠ 0, speed increases if va > 0 and decreases if va < 0; the sign of acceleration alone does not determine speeding up.

What are the energies of a satellite in a circular orbit, and how do they change between circular orbits?

For m ≪ M orbiting outside a fixed spherically symmetric central body, with U(∞) = 0: U = −GMm/r, K = GMm/(2r) and E = −GMm/(2r). Comparing circular orbits, larger r gives greater U and E but smaller K and speed. Raising the final circular orbit requires a net energy input; this comparison does not describe the speed change at every instant of a maneuver.

How is work defined, and what does the work–energy theorem state?

For constant force, W = F·Δr = Fd cosθ. In general, W = ∫F·dr along the path; in one dimension this is ∫Fx dx, the signed force–position area. For a Newtonian particle of constant mass, W_net = ΔK = ½m(vf² − vi²). Work is measured in joules, with 1 J = 1 N·m.

Why does a Kepler orbit sweep equal areas in equal times?

The gravitational force toward a fixed central mass exerts zero torque about that center, so orbital angular momentum L is conserved. The areal rate is dA/dt = L/(2m), using the magnitude of L. In an elliptical orbit, speed is greatest at periapsis and smallest at apoapsis; there the velocity is transverse, so rp vp = ra va.

How does an elevator scale reading depend on vertical acceleration?

With upward positive, a passenger sharing the elevator’s acceleration ay and subject only to gravity and the scale force has N = m(g + ay). Thus ay = 0 gives N = mg, ay > 0 gives N > mg, and −g < ay < 0 gives N < mg. Free fall gives N = 0. The contact model requires N ≥ 0; it cannot predict a negative normal force.

What are the acceleration and tension of an ideal Atwood machine with m2 > m1 > 0?

For a massless inextensible rope over a massless frictionless pulley in uniform gravity, a = (m2 − m1)g/(m1 + m2): m2 accelerates downward and m1 upward. The rope tension is T = 2m1m2g/(m1 + m2), so m1g < T < m2g.

What are Hooke’s law, spring energy and the stiffnesses of ideal spring combinations?

For displacement x from the unstretched position, Fs = −kx and Us = ½kx² with Us(0) = 0. Us increases by the work done against the spring, not by the signed area under Fs. For massless linear springs, equal extensions in parallel give k_eq = k1 + k2; equal forces in series give 1/k_eq = 1/k1 + 1/k2.

How is the ideal ballistic-pendulum problem solved in two stages?

A horizontal bullet of mass m embeds in an initially stationary pendulum mass M. With negligible external horizontal impulse, mv0 = (m + M)v_after. Model the combined mass as a particle in uniform gravity g and neglect swing losses: ½(m + M)v_after² = (m + M)gh, where h is its rise. Thus v0 = ((m + M)/m)√(2gh). Kinetic energy is not conserved during embedding.

How do relative velocities add, and what sets the crossing time of an ideal river boat?

In classical mechanics using common nonrotating axes, v_A/C = v_A/B + v_B/C as vectors. If a boat maintains speed v_bw relative to water and points straight across parallel banks, while uniform current v_wg runs along the banks, a river of width W is crossed in t = W/v_bw. Downstream drift is Δx = v_wg t; current independence of t holds only under these assumptions.

What is the gravitational field inside and outside a uniform solid sphere?

For mass M, radius R and distance r from the center, the field points inward for r > 0. Its magnitude is g(r) = GM/r² for r ≥ R and g(r) = GMr/R³ for 0 ≤ r 0; g(0) = 0. A spherically symmetric hollow shell produces zero field in its empty interior, but does not shield fields from other masses.

What are momentum and impulse, and why distinguish average from peak impact force?

For constant mass, p = mv and J = ∫F_net dt = Δp = F_avg Δt, all vector relations. An impulse component equals the signed area under the corresponding force–time graph. For fixed Δp, increasing impact time reduces the magnitude of the average net force. Peak force also depends on the time profile and is not determined by that average alone.

What is the contact force between two adjacent blocks pushed along a frictionless table?

Let m1 be the left block, m2 the right block, and F > 0 push m1 to the right. Both accelerate at a = F/(m1 + m2). The only horizontal force on m2 is the contact force, so F_1on2 = m2F/(m1 + m2), directed right. The force on m1 from m2 is equal and opposite. Pushing from the other side swaps the mass factor.

How does a body dropped from rest move under linear drag F_drag = −bv?

Take downward as positive, with constant m, g and b > 0 and negligible buoyancy. Then m dv/dt = mg − bv. The terminal velocity is vT = mg/b, the time constant is τ = m/b, and v(t) = vT(1 − e^(−t/τ)). Acceleration is a(t) = g e^(−t/τ). This assumes the linear-drag regime.

How do static and kinetic friction differ, and what is the ideal angle of repose?

In the simple friction model, 0 ≤ fs ≤ μsN while surfaces do not slide; during sliding, fk = μkN opposes relative motion. μk is often below μs, but this is not a universal identity, nor is fk constant if N changes. For a block on an incline acted on only by gravity, normal force and friction, impending slip gives tanθ_max = μs.

How does the choice of energy bookkeeping affect gravity, springs and friction?

A particle-only model can count gravity and spring forces as work: Wg + Ws + W_other = ΔK. Alternatively, represent their work by interaction energies in U and write Δ(K + U) = W_other, without counting those forces twice. If friction heats bodies within the chosen system, include internal energy: for an isolated system, ΔK + ΔU + ΔE_internal = 0.

What four equations describe one-dimensional motion with constant acceleration?

For elapsed time t, signed displacement Δx, initial velocity v0 and constant acceleration a: v = v0 + at; Δx = v0t + ½at²; v² = v0² + 2aΔx; Δx = ½(v0 + v)t. These respectively omit Δx, v, t and a.

How are kinetic energy and momentum magnitude related for a nonrelativistic particle?

For m > 0, K = p²/(2m) and p = √(2mK), where p is momentum magnitude. At equal K, a cart of mass 4m has twice the momentum magnitude of one of mass m. At equal p, the mass-m cart has four times the kinetic energy of the mass-4m cart.

How are center-of-mass motion and total momentum related?

For a fixed set of constant masses with total M, r_cm = Σmi ri/M and v_cm = Σmi vi/M = P_total/M; these are vector relations. In Newtonian mechanics with cancelling internal forces, F_ext,net = M a_cm. If the net external force is zero, v_cm remains constant even during internal collisions or explosions; individual external forces need not each be absent.

What happens in a one-dimensional elastic collision with a stationary target?

With negligible external impulse and initial velocity v0 > 0 for mass m1, v1f = ((m1 − m2)/(m1 + m2))v0 and v2f = (2m1/(m1 + m2))v0. Equal masses exchange velocities. For m1 ≪ m2, v1f ≈ −v0 and v2f ≈ 0; for m1 ≫ m2, v1f ≈ v0 and v2f ≈ 2v0.

How does a one-dimensional potential-energy curve describe force, equilibrium and turning points?

Fx = −dU/dx and ΔU = −∫Fx dx. Equilibrium requires U′ = 0. At such a point, U″ > 0 gives stable equilibrium and U″ < 0 unstable equilibrium; U″ = 0 alone is inconclusive. A locally constant potential gives neutral equilibrium. A turning point has K = 0 and E = U, although not every E = U point is necessarily a reversal.

What minimum release height lets a particle slide around an inside vertical loop without losing contact?

For a frictionless circular track of radius R, a particle released from rest at height h above the bottom needs v_top² ≥ gR at the top. Energy gives mgh = mg(2R) + ½mv_top², so h_min = 5R/2. This assumes sliding without rotational kinetic energy, uniform gravity and no retaining constraint that can pull the particle toward the track.

What are the ideal speeds for a flat friction-limited turn and a frictionless banked turn?

For a point-mass car making a steady circular turn of radius R, with no other horizontal forces, a flat road gives v_max = √(μs gR). For a frictionless bank angle θ above horizontal, N sinθ = mv²/R and N cosθ = mg, giving v = √(gR tanθ). These are ideal tire-contact models, not universal road-speed limits.

AP Physics 1 & C: Kinematics, Newton's Laws, Work-Energy Theorem & Linear Momentum

32 algebra- and calculus-based mechanics review cards on motion, forces, work and energy, collisions, gravitation, and experimental graphs. Assumes algebra, trigonometry, vectors and basic calculus.

Level: basic
Languages: en → en
Cards: 32
Markdown: AP Physics 1 & C: Kinematics, Newton's Laws, Work-Energy Theorem & Linear Momentum (.md)

Flashcards (32)

#	Front (Term / Question)	Back (Definition / Answer)
1	How do gravity, circular-orbit speed, period and escape speed relate outside a spherically symmetric central body?	For a test mass m ≪ M outside an isolated spherically symmetric mass M, F = GMm/r² and g = GM/r². Circular orbits have v_orb = √(GM/r) and T² = 4π²r³/(GM). Escape from radius R under gravity alone requires v_esc = √(2GM/R), equal to √2 times the circular speed at that same radius. Ignore atmospheric drag and source rotation.
2	What are the apex properties, flight time, height and range of an ideal level-ground projectile?	With uniform downward g, no air resistance, launch speed v0 and angle 0 < θ < 90°, the apex has vx = v0 cosθ, vy = 0, ax = 0 and ay = −g. Returning to launch height takes T = 2v0 sinθ/g. Height above launch is H = v0² sin²θ/(2g), and range is R = v0² sin(2θ)/g. At fixed v0, range is greatest at 45°; complementary angles give equal ranges.
3	Where is a falling block’s speed greatest while compressing an ideal vertical spring?	For mass m dropped from rest a height h ≥ 0 above an unstretched massless spring of stiffness k, neglect drag and dissipation. At downward compression y, a = g − ky/m, so speed peaks at y = mg/k. From mg(h + y) = ½mv² + ½ky², v_max = √(2gh + mg²/k). Maximum speed occurs after first contact, while the spring remains in its linear range.
4	How is momentum conservation resolved for a two-dimensional glancing collision?	Suppose m1 initially moves along +x at speed v1i, m2 is stationary, and external impulse is negligible. With final speeds v1f and v2f at angles θ1 above and θ2 below +x: m1v1i = m1v1f cosθ1 + m2v2f cosθ2, and 0 = m1v1f sinθ1 − m2v2f sinθ2. These momentum equations do not require an elastic collision.
5	What is centripetal acceleration, and what speed keeps an unrestrained particle in contact at the top of an inside loop?	The inward radial acceleration is ac = v²/r = ω²r; “centripetal force” names the net inward force, not an extra interaction. At the top of a circular inside track of radius R, N + mg = mv²/R. Requiring N ≥ 0 gives v_min = √(gR). This contact condition assumes gravity and the track normal are the only radial forces.
6	What do Newton’s three laws say in an inertial frame, and what is a third-law pair?	Zero net force gives constant velocity. For a particle of constant mass, ΣF = dp/dt = ma, with vector quantities. For an interaction between A and B, F_AonB = −F_BonA. These equal and opposite forces act on different bodies. Weight and normal force on the same book are not a third-law pair.
7	How can linearizing a physics relation reveal a constant from a fitted slope?	For a small-angle simple pendulum, T² = (4π²/g)L. Plot T² against L: the ideal slope S gives g = 4π²/S. Estimate S from a fit; if using a graph, use well-separated points on the fitted line. The ideal intercept is zero, but do not force measured data through zero without justification. For constant acceleration from rest, plotting v² against Δx gives slope 2a.
8	What are average and instantaneous mechanical power?	Average power for work W over time Δt is P_avg = W/Δt. Instantaneous power is P = dW/dt. For a force acting on a particle with velocity v, P = F·v = Fv cosθ. Power measures the rate of energy transfer by work; 1 W = 1 J/s.
9	What forces and acceleration apply to a block sliding down a fixed rough incline?	For slope angle θ above horizontal, gravity has components mg sinθ down the slope and mg cosθ into it. With no other applied force and no normal acceleration, N = mg cosθ. In the kinetic-friction model, fk = μkN acts uphill. Taking downhill as positive, a = g(sinθ − μk cosθ) while the block slides downhill; a may be negative.
10	How do momentum conservation and collision types relate?	Total momentum is constant when net external force is zero; negligible external impulse suffices across a short collision. Elastic collisions conserve kinetic energy. Dissipative inelastic collisions reduce it; superelastic collisions increase it by releasing internal energy. In the point-mass model, perfectly inelastic objects share a final velocity, minimizing final translational kinetic energy for the given total momentum and masses.
11	How do position, velocity and acceleration relate through calculus and motion graphs?	In one dimension, v = dx/dt and a = dv/dt. Signed areas give Δx = ∫v dt and Δv = ∫a dt over the time interval. Where v ≠ 0, speed increases if va > 0 and decreases if va < 0; the sign of acceleration alone does not determine speeding up.
12	What are the energies of a satellite in a circular orbit, and how do they change between circular orbits?	For m ≪ M orbiting outside a fixed spherically symmetric central body, with U(∞) = 0: U = −GMm/r, K = GMm/(2r) and E = −GMm/(2r). Comparing circular orbits, larger r gives greater U and E but smaller K and speed. Raising the final circular orbit requires a net energy input; this comparison does not describe the speed change at every instant of a maneuver.
13	How is work defined, and what does the work–energy theorem state?	For constant force, W = F·Δr = Fd cosθ. In general, W = ∫F·dr along the path; in one dimension this is ∫Fx dx, the signed force–position area. For a Newtonian particle of constant mass, W_net = ΔK = ½m(vf² − vi²). Work is measured in joules, with 1 J = 1 N·m.
14	Why does a Kepler orbit sweep equal areas in equal times?	The gravitational force toward a fixed central mass exerts zero torque about that center, so orbital angular momentum L is conserved. The areal rate is dA/dt = L/(2m), using the magnitude of L. In an elliptical orbit, speed is greatest at periapsis and smallest at apoapsis; there the velocity is transverse, so rp vp = ra va.
15	How does an elevator scale reading depend on vertical acceleration?	With upward positive, a passenger sharing the elevator’s acceleration ay and subject only to gravity and the scale force has N = m(g + ay). Thus ay = 0 gives N = mg, ay > 0 gives N > mg, and −g < ay < 0 gives N < mg. Free fall gives N = 0. The contact model requires N ≥ 0; it cannot predict a negative normal force.
16	What are the acceleration and tension of an ideal Atwood machine with m2 > m1 > 0?	For a massless inextensible rope over a massless frictionless pulley in uniform gravity, a = (m2 − m1)g/(m1 + m2): m2 accelerates downward and m1 upward. The rope tension is T = 2m1m2g/(m1 + m2), so m1g < T < m2g.
17	What are Hooke’s law, spring energy and the stiffnesses of ideal spring combinations?	For displacement x from the unstretched position, Fs = −kx and Us = ½kx² with Us(0) = 0. Us increases by the work done against the spring, not by the signed area under Fs. For massless linear springs, equal extensions in parallel give k_eq = k1 + k2; equal forces in series give 1/k_eq = 1/k1 + 1/k2.
18	How is the ideal ballistic-pendulum problem solved in two stages?	A horizontal bullet of mass m embeds in an initially stationary pendulum mass M. With negligible external horizontal impulse, mv0 = (m + M)v_after. Model the combined mass as a particle in uniform gravity g and neglect swing losses: ½(m + M)v_after² = (m + M)gh, where h is its rise. Thus v0 = ((m + M)/m)√(2gh). Kinetic energy is not conserved during embedding.
19	How do relative velocities add, and what sets the crossing time of an ideal river boat?	In classical mechanics using common nonrotating axes, v_A/C = v_A/B + v_B/C as vectors. If a boat maintains speed v_bw relative to water and points straight across parallel banks, while uniform current v_wg runs along the banks, a river of width W is crossed in t = W/v_bw. Downstream drift is Δx = v_wg t; current independence of t holds only under these assumptions.
20	What is the gravitational field inside and outside a uniform solid sphere?	For mass M, radius R and distance r from the center, the field points inward for r > 0. Its magnitude is g(r) = GM/r² for r ≥ R and g(r) = GMr/R³ for 0 ≤ r < R. Inside, M_enclosed = M(r/R)³ and g = GM_enclosed/r² for r > 0; g(0) = 0. A spherically symmetric hollow shell produces zero field in its empty interior, but does not shield fields from other masses.
21	What are momentum and impulse, and why distinguish average from peak impact force?	For constant mass, p = mv and J = ∫F_net dt = Δp = F_avg Δt, all vector relations. An impulse component equals the signed area under the corresponding force–time graph. For fixed Δp, increasing impact time reduces the magnitude of the average net force. Peak force also depends on the time profile and is not determined by that average alone.
22	What is the contact force between two adjacent blocks pushed along a frictionless table?	Let m1 be the left block, m2 the right block, and F > 0 push m1 to the right. Both accelerate at a = F/(m1 + m2). The only horizontal force on m2 is the contact force, so F_1on2 = m2F/(m1 + m2), directed right. The force on m1 from m2 is equal and opposite. Pushing from the other side swaps the mass factor.
23	How does a body dropped from rest move under linear drag F_drag = −bv?	Take downward as positive, with constant m, g and b > 0 and negligible buoyancy. Then m dv/dt = mg − bv. The terminal velocity is vT = mg/b, the time constant is τ = m/b, and v(t) = vT(1 − e^(−t/τ)). Acceleration is a(t) = g e^(−t/τ). This assumes the linear-drag regime.
24	How do static and kinetic friction differ, and what is the ideal angle of repose?	In the simple friction model, 0 ≤ fs ≤ μsN while surfaces do not slide; during sliding, fk = μkN opposes relative motion. μk is often below μs, but this is not a universal identity, nor is fk constant if N changes. For a block on an incline acted on only by gravity, normal force and friction, impending slip gives tanθ_max = μs.
25	How does the choice of energy bookkeeping affect gravity, springs and friction?	A particle-only model can count gravity and spring forces as work: Wg + Ws + W_other = ΔK. Alternatively, represent their work by interaction energies in U and write Δ(K + U) = W_other, without counting those forces twice. If friction heats bodies within the chosen system, include internal energy: for an isolated system, ΔK + ΔU + ΔE_internal = 0.
26	What four equations describe one-dimensional motion with constant acceleration?	For elapsed time t, signed displacement Δx, initial velocity v0 and constant acceleration a: v = v0 + at; Δx = v0t + ½at²; v² = v0² + 2aΔx; Δx = ½(v0 + v)t. These respectively omit Δx, v, t and a.
27	How are kinetic energy and momentum magnitude related for a nonrelativistic particle?	For m > 0, K = p²/(2m) and p = √(2mK), where p is momentum magnitude. At equal K, a cart of mass 4m has twice the momentum magnitude of one of mass m. At equal p, the mass-m cart has four times the kinetic energy of the mass-4m cart.
28	How are center-of-mass motion and total momentum related?	For a fixed set of constant masses with total M, r_cm = Σmi ri/M and v_cm = Σmi vi/M = P_total/M; these are vector relations. In Newtonian mechanics with cancelling internal forces, F_ext,net = M a_cm. If the net external force is zero, v_cm remains constant even during internal collisions or explosions; individual external forces need not each be absent.
29	What happens in a one-dimensional elastic collision with a stationary target?	With negligible external impulse and initial velocity v0 > 0 for mass m1, v1f = ((m1 − m2)/(m1 + m2))v0 and v2f = (2m1/(m1 + m2))v0. Equal masses exchange velocities. For m1 ≪ m2, v1f ≈ −v0 and v2f ≈ 0; for m1 ≫ m2, v1f ≈ v0 and v2f ≈ 2v0.
30	How does a one-dimensional potential-energy curve describe force, equilibrium and turning points?	Fx = −dU/dx and ΔU = −∫Fx dx. Equilibrium requires U′ = 0. At such a point, U″ > 0 gives stable equilibrium and U″ < 0 unstable equilibrium; U″ = 0 alone is inconclusive. A locally constant potential gives neutral equilibrium. A turning point has K = 0 and E = U, although not every E = U point is necessarily a reversal.
31	What minimum release height lets a particle slide around an inside vertical loop without losing contact?	For a frictionless circular track of radius R, a particle released from rest at height h above the bottom needs v_top² ≥ gR at the top. Energy gives mgh = mg(2R) + ½mv_top², so h_min = 5R/2. This assumes sliding without rotational kinetic energy, uniform gravity and no retaining constraint that can pull the particle toward the track.
32	What are the ideal speeds for a flat friction-limited turn and a frictionless banked turn?	For a point-mass car making a steady circular turn of radius R, with no other horizontal forces, a flat road gives v_max = √(μs gR). For a frictionless bank angle θ above horizontal, N sinθ = mv²/R and N cosθ = mg, giving v = √(gR tanθ). These are ideal tire-contact models, not universal road-speed limits.