Section 1.8: Problem-Solving Strategies (Kinematics)

Effective kinematics problem-solving requires a step-by-step approach. Breaking problems into knowns and unknowns, choosing appropriate equations, and carefully analyzing vectors are essential.

Step 1: Analyze the problem
  • Identify all known quantities (initial velocity \(v_0\), final velocity \(v_f\), acceleration \(a\), displacement \(d\), time \(t\)).
  • Identify the unknown quantity you need to find.
  • Draw a clear diagram if motion is in 1D or 2D.
Step 2: Choose the right equation

Use the kinematic equations suitable for the problem. For uniform acceleration in one dimension, you may use:

  • \(v_f = v_0 + a t\)
  • \(d = v_0 t + \frac{1}{2} a t^2\)
  • \(v_f^2 = v_0^2 + 2 a d\)
  • \(d = \frac{v_0 + v_f}{2} t\)
Step 3: Solve algebraically
  • Substitute known values carefully.
  • Keep track of units (meters, seconds, m/s²).
  • Check sign conventions for direction.
Step 4: Verify your solution
  • Check if the answer is physically reasonable (e.g., displacement matches motion diagram).
  • Verify units.
  • Ensure velocity, acceleration, and time make sense relative to each other.

Example 1

A car starts from rest and accelerates at 3 m/s² for 6 s. Find final velocity and displacement using problem-solving strategy.

Step 1: Known: \(v_0=0\), \(a=3\), \(t=6\). Unknown: \(v_f, d\)

Step 2: Use \(v_f = v_0 + a t = 0 + 3*6 = 18 \text{ m/s}\)

Use \(d = v_0 t + 0.5 a t^2 = 0 + 0.5*3*36 = 54 \text{ m}\)

Example 2

A runner accelerates from 4 m/s to 12 m/s over 20 m. Find acceleration using the right equation.

Known: \(v_0=4\), \(v_f=12\), \(d=20\), unknown: \(a\)

Use \(v_f^2 = v_0^2 + 2 a d \Rightarrow 12^2 = 4^2 + 2 a*20\)

\(144 -16 = 40a \Rightarrow a = 128/40 = 3.2 \text{ m/s²}\)

Practice Problems

  1. A car accelerates from 0 to 20 m/s in 8 s. Find acceleration and displacement.
  2. A particle moves 50 m in 5 s from rest under constant acceleration. Find final velocity and acceleration.
  3. A cyclist traveling at 6 m/s accelerates at 1 m/s² for 10 s. Find final velocity and displacement.
  4. A runner decelerates from 12 m/s to rest over 30 m. Find deceleration and time taken.
  5. A train accelerates from 10 m/s to 25 m/s over 200 m. Find acceleration.
  6. A car starts from rest and covers 100 m in 5 s with constant acceleration. Find final velocity.
  7. An object slows from 15 m/s to 5 m/s in 4 s. Find acceleration and displacement.
  8. A car moves with uniform velocity of 20 m/s for 15 s. Find displacement.
  9. A particle moves with v=2t m/s. Find displacement after 6 s using integration (graphical approximation allowed).
  10. A body accelerates at 3 m/s² from rest. How long to reach 30 m/s and how far has it moved?
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