Kinematics and Motion

Grade 11 · Science · Study Guide

Analyze one-dimensional and projectile motion using kinematic equations, velocity, acceleration, and graphical methods.

Kinematics is the branch of physics that describes motion without explaining its causes. The four key quantities are displacement, velocity, acceleration, and time. Displacement (delta x) is the change in position — a vector quantity with both magnitude and direction — while distance is scalar. Average velocity is v_avg = delta x divided by delta t, while instantaneous velocity is the velocity at a specific moment.

Uniform acceleration problems use the four kinematic equations: (1) v = v_0 + at, (2) delta x = v_0 t + one-half a t^2, (3) v^2 = v_0^2 + 2a delta x, and (4) delta x = one-half (v + v_0) t. Free fall is a special case where a = 9.8 m/s^2 downward (taking downward as positive, or -9.8 if upward is positive). Always identify known and unknown variables before selecting an equation.

Position-time graphs have slope equal to velocity. Velocity-time graphs have slope equal to acceleration, and the area under the curve equals displacement. These graphical interpretations are tested frequently and deeply connect to calculus concepts.

Projectile motion combines independent horizontal (constant velocity) and vertical (constant acceleration due to gravity) components. Both components share the same time of flight. At maximum height, the vertical velocity component is zero. Range is maximized at a launch angle of 45 degrees.

Key terms

Displacement
The vector change in position from start to finish; delta x = x_final minus x_initial.
Acceleration
The rate of change of velocity; a = delta v divided by delta t, measured in m/s^2.
Free Fall
Motion under gravity alone, with constant downward acceleration g approximately 9.8 m/s^2.
Projectile Motion
Two-dimensional motion with constant horizontal velocity and constant vertical acceleration due to gravity.
Velocity-Time Graph
A graph where slope equals acceleration and area under the curve equals displacement.

Flashcards

Kinematic Equation 1
v = v_0 + at; final velocity from initial velocity, acceleration, and time.
Kinematic Equation 2
delta x = v_0 t + one-half a t^2; displacement from initial velocity, time, and acceleration.
Kinematic Equation 3
v^2 = v_0^2 + 2a delta x; relates velocity and displacement without time.
Scalar vs. Vector
Scalars have magnitude only (speed, distance); vectors have magnitude and direction (velocity, displacement).
Average Velocity
v_avg = delta x divided by delta t; total displacement divided by total time.
Instantaneous Velocity
The velocity at a single point in time; the slope of the tangent on a position-time graph.
g (gravitational acceleration)
9.8 m/s^2 directed downward near Earth's surface; used in free-fall and projectile calculations.
Range of a Projectile
Horizontal distance traveled; maximized at 45 degree launch angle for flat terrain.

Questions

  1. A car accelerates from rest at 3 m/s^2 for 5 seconds. What is its final velocity?

    1. 5 m/s
    2. 10 m/s
    3. 15 m/s
    4. 25 m/s
    Show answer

    Answer: 15 m/s

    v = v_0 + at = 0 + (3)(5) = 15 m/s.

  2. On a velocity-time graph, the ___ under the curve equals the displacement.

    Show answer

    Answer: area

    The area between the velocity-time graph and the time axis equals displacement (delta x = v times t for constant v, or the integral of v(t)).

  3. A ball is dropped from rest. After 3 seconds (g = 9.8 m/s^2), how far has it fallen?

    1. 14.7 m
    2. 29.4 m
    3. 44.1 m
    4. 88.2 m
    Show answer

    Answer: 44.1 m

    delta x = v_0 t + one-half a t^2 = 0 + one-half (9.8)(3^2) = one-half (9.8)(9) = 44.1 m.

  4. At the maximum height of a projectile's trajectory, its vertical velocity component is zero.

    Show answer

    Answer: True

    At peak height the projectile momentarily stops moving vertically before gravity pulls it back down, so v_y = 0.

  5. Which kinematic equation would you use when time is unknown and you need to relate velocity and displacement?

    1. v = v_0 + at
    2. delta x = v_0 t + one-half a t^2
    3. v^2 = v_0^2 + 2a delta x
    4. delta x = one-half (v + v_0) t
    Show answer

    Answer: v^2 = v_0^2 + 2a delta x

    v^2 = v_0^2 + 2a delta x is the only kinematic equation that relates velocity and displacement without requiring time.

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