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Flashcards · AP® Physics C: Mechanics · Unit 1

Unit 1: Kinematics flashcards

39 cards on topics 1.1 to 1.5. Pick a topic or study the whole unit.

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1.1 Scalars and Vectors

What is the difference between a scalar and a vector?
A scalar has magnitude only (distance, speed, time, mass). A vector has magnitude and direction (position, displacement, velocity, acceleration). A vector is drawn as an arrow pointing its way, with length proportional to its magnitude on a stated scale.
How do you write a vector in unit-vector notation, and find its magnitude?
A = A_x î + A_y ĵ + A_z k̂, where the components are signed scalars and î, ĵ, k̂ each have magnitude 1. The magnitude is |A| = √(A_x² + A_y² + A_z²), which is never negative.
How do you resolve a vector into components?
With θ measured from +x towards +y: A_x = |A| cos θ and A_y = |A| sin θ. If the angle is measured from the y-axis, sine and cosine swap. Keep the sign of any component pointing left, down, south or west.
How do you find the direction of a vector from its components?
tan θ = A_y / A_x, measured from +x. A calculator's tan⁻¹ only returns angles in two quadrants, so sketch the vector and check which quadrant it is in before giving θ.
How do you add two vectors?
Add component by component: R_x = A_x + B_x, R_y = A_y + B_y, R_z = A_z + B_z. Never add the magnitudes. The size of the resultant lies between |A| − |B| and |A| + |B| (for |A| ≥ |B|).
What is the unit vector r̂ along a position vector r?
r̂ = r / |r|. It points the same way as r, has magnitude exactly 1, and has no unit.
How do you find the size of a displacement from two position vectors?
Subtract the vectors first, Δr = r − r₀, then take the magnitude. |r₂| − |r₁| is not |Δr| unless the two vectors are parallel.
Is distance always equal to the magnitude of displacement?
No. They are equal only for motion in one direction along a straight line. On a path that bends or turns back, the distance travelled is larger than |Δr|.

1.2 Displacement, Velocity, and Acceleration

How are position, velocity and acceleration related by calculus?
Differentiate down: v_x = dx/dt (m/s) and a_x = dv_x/dt = d²x/dt² (m/s²). Integrate up: v_x(t) = v_x0 + ∫₀ᵗ a_x dt and x(t) = x₀ + ∫₀ᵗ v_x dt, using the initial conditions.
What is the difference between average and instantaneous velocity?
Average velocity uses only the initial and final states: v_avg = Δx/Δt. Instantaneous velocity is the limit as Δt → 0, the derivative dx/dt, the tangent slope of the x–t graph.
When can you use the constant-acceleration equations?
Only when a_x is constant: v_x = v_x0 + a_xt; x = x₀ + v_x0t + ½a_xt²; v_x² = v_x0² + 2a_x(x − x₀). If a_x depends on t, integrate instead.
How do you find distance travelled, rather than displacement, from v_x(t)?
Displacement is ∫v_x dt (signed). Distance is ∫|v_x| dt: split the interval where v_x = 0 and add the sizes of each part.
Does a negative acceleration mean the object is slowing down?
Not necessarily. An object slows down when v_x and a_x have opposite signs and speeds up when they have the same sign. A negative a_x with a negative v_x means speeding up.
If v_x = 0 at an instant, is a_x also zero?
No. Velocity and acceleration are separate. At a turning point v_x = 0 while a_x can be large, for example a ball at the top of a throw still has a = g downward.
What is the commonest mistake when integrating acceleration to find velocity?
Forgetting the constant of integration. An indefinite integral needs the initial condition: v_x(t) = v_x0 + ∫₀ᵗ a_x dt, and x(t) = x₀ + ∫₀ᵗ v_x dt.
x(t) = 4.0t² − t³ (m). What is v_x at t = 2.0 s?
v_x = dx/dt = 8.0t − 3t² (m/s). At t = 2.0 s, v_x = 16 − 12 = 4.0 m/s.

1.3 Representing Motion

How do you read velocity and acceleration from an x–t graph?
Velocity is the slope of the tangent to the x–t graph, not its height. Acceleration is the curvature: concave up means a_x is positive, concave down means a_x is negative.
What does the area under a v_x–t graph or an a_x–t graph give?
Signed area under v_x–t gives the displacement Δx. Signed area under a_x–t gives the change in velocity Δv_x, so add v_x0 to get v_x.
What does a motion diagram show about acceleration?
Dots are placed at equal time steps. Growing gaps mean speeding up, shrinking gaps mean slowing down, and equal changes in the gaps mean constant acceleration.
Which constant-acceleration equation do you choose?
Pick the one missing the quantity you do not know or need: v_x = v_x0 + a_xt has no x; x = x₀ + v_x0t + ½a_xt² has no v_x; v_x² = v_x0² + 2a_x(x − x₀) has no t.
What is the acceleration of a ball at the top of a vertical throw?
Still g ≈ 9.8 m/s² downward. Only the velocity is zero there. With +y up, a_y = −9.8 m/s² at every point of the flight, ignoring air resistance.
How do you handle motion that has two stages of different acceleration?
Restart the equations at each change of acceleration, using the final values of one stage as the initial values of the next. One equation cannot span both stages.
Can an x–t graph have a sharp corner?
Not for a real object. Velocity cannot jump instantly, so the slope of x–t changes smoothly and the graph bends without corners.
How do you answer a factor-of-change question in kinematics?
Write the result as a proportion first. For example h = v₀²/(2g) gives h ∝ v₀², so doubling the launch speed gives four times the height. A drop from rest gives t ∝ √h.

1.4 Reference Frames and Relative Motion

What is a reference frame?
An observer's origin, axes, clock and state of motion. The frame decides both the size and direction of measured positions and velocities, so state it before any number, for example "+x east, measured from the ground".
How do you convert a velocity from one frame to another?
v_PA = v_PB + v_BA, where v_PA means the velocity of P measured in frame A. The inner letters match and drop out. Add the velocities as vectors, using components when they are not parallel.
How is v_BA related to v_AB?
v_BA = −v_AB. Swapping the order of the subscripts reverses the vector, so the minus sign carries direction.
Do inertial observers measure the same acceleration?
Yes. Differentiating the velocity link gives a_PA = a_PB + a_BA, and if frame B moves at constant velocity relative to A, a_BA = 0, so a_PA = a_PB.
How do you aim to travel straight across a crosswind?
Head into the wind at angle θ from the desired track with sin θ = w/u, where w is the wind (or current) speed and u the airspeed. The ground speed is then √(u² − w²). This needs w smaller than u.
What is the difference between heading and track?
Heading is the direction the craft points relative to the air or water. Track is its actual path over the ground. They differ whenever the medium moves sideways relative to the ground.
A plane flies north at 80 m/s relative to the air; the wind blows east at 60 m/s. What is its ground velocity?
v_ground = v_plane,air + v_air,ground = (60 î + 80 ĵ) m/s. Speed √(60² + 80²) = 100 m/s, at 36.9° east of north.

1.5 Motion in Two or Three Dimensions

How do you analyse motion in two dimensions?
Treat it as two independent one-dimensional motions. Write r(t) = x(t) î + y(t) ĵ, fill a table of initial position, velocity and acceleration for x and y, and solve each axis separately with the same time t.
How do you find velocity and acceleration from r(t) in two dimensions?
Differentiate each component: v_x = dx/dt, v_y = dy/dt, a_x = dv_x/dt, a_y = dv_y/dt. Velocity is tangent to the path; acceleration need not be parallel to velocity.
What are the equations for a projectile launched from the origin?
With +y up and no air resistance: x = v₀ cos θ · t and y = v₀ sin θ · t − ½gt². Horizontally a_x = 0; vertically a_y = −g.
When can you use the range formula R = v₀² sin 2θ / g?
Only when launch and landing heights are equal and air resistance is ignored. From a cliff or onto a slope, solve y(t) = landing height for t instead.
How long does an object launched horizontally from height h take to land?
t = √(2h/g). The fall time does not depend on the horizontal speed, because horizontal motion does not affect vertical motion.
What are the speed and acceleration of a projectile at the top of its path?
Only v_y = 0, so the speed equals the horizontal component v₀ cos θ. The acceleration is still g downward.
What shape is a projectile's path?
A parabola: y = x tan θ − gx² / (2v₀² cos²θ), found by eliminating t between x(t) and y(t).
r(t) = (2.0t²) î + (3.0t) ĵ (m). What is the speed at t = 1.0 s?
v = dr/dt = (4.0t) î + 3.0 ĵ m/s. At t = 1.0 s, v = 4.0 î + 3.0 ĵ, so the speed is √(16 + 9) = 5.0 m/s.

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