Section 1.1: Motion in One Dimension

Motion in one dimension deals with objects moving along a straight line. Understanding this requires distinguishing between distance and displacement, as well as speed, velocity, and acceleration.

Distance vs Displacement:
  • Distance: Total path traveled (scalar, always positive).
  • Displacement: Change in position (vector, can be negative).
Speed vs Velocity:
  • Speed: Rate of change of distance (scalar).
  • Velocity: Rate of change of displacement (vector).
Acceleration: Rate of change of velocity with time. Positive or negative (deceleration).

The Four Kinematic Equations (Constant Acceleration)

Let vi = initial velocity, vf = final velocity, a = acceleration, d = displacement, t = time.

  • \(v_f = v_i + a t\)
  • \(d = v_i t + \frac{1}{2} a t^2\)
  • \(v_f^2 = v_i^2 + 2 a d\)
  • \(d = \frac{v_i + v_f}{2} t\)
Example: A car starts from rest (\(v_i = 0\)) and accelerates at 3 m/s² for 5 s. Find the final velocity and displacement.

Final velocity: \(v_f = 0 + (3)(5) = 15 \text{ m/s}\)

Displacement: \(d = 0*5 + 0.5*(3)*(5^2) = 37.5 \text{ m}\)

Example 1

A car starts from rest and accelerates at 2 m/s² for 5 s. Find its final velocity and displacement.

Final velocity: \(v = 0 + 2*5 = 10\text{ m/s}\)

Displacement: \(s = 0 + 0.5*2*25 = 25\text{ m}\)

Example 2

A car moving at 20 m/s slows down at 4 m/s² until it stops. Find the stopping distance.

\(v^2 = u^2 + 2as \Rightarrow 0 = 20^2 + 2(-4)s \Rightarrow s = 100 \text{ m}\)

Horizontal Kinematics Problems

  1. A drag racer accelerates from 0 to 100 m/s in 4 s. Assuming constant acceleration, how far did the car travel?
  2. A train traveling at 25 m/s begins to slow down with a constant acceleration of -2 m/s². How long does it take for the train to come to a stop?
  3. A car accelerates from 10 m/s to 30 m/s over a distance of 120 m. What is its acceleration?
  4. An object, starting from rest, travels 50 m in 8 s. Assuming constant acceleration, what is its final velocity?
  5. A cyclist with a velocity of 5 m/s increases their speed with an acceleration of 1.5 m/s² over a distance of 20 m. What is their final velocity?
  6. A runner traveling at 8 m/s begins to decelerate at a constant rate of 1.2 m/s². How far will they travel before coming to a stop?
  7. An object accelerates from 15 m/s to 25 m/s over a period of 4 s. What is the displacement during this time?
  8. A car traveling at 20 m/s hits a patch of ice and slides for 6 s before stopping. Assuming constant deceleration, what is the distance it traveled on the ice?
  9. An object accelerates from rest at 3 m/s² for 10 s. What is its final velocity and how far has it traveled?
  10. A motorcycle traveling at 30 m/s applies the brakes and stops in 5 s. What is the motorcycle's constant acceleration and the distance traveled while braking?
  11. An object moves with a constant acceleration of 2 m/s². If its initial velocity is 4 m/s and its final velocity is 16 m/s, what is its displacement?
  12. A car accelerates from 5 m/s to 15 m/s over a distance of 60 m. How much time did this take?
  13. A runner starts from rest and accelerates at a constant rate. After traveling 50 m, their velocity is 10 m/s. What is their acceleration?
  14. An object with a velocity of 12 m/s slows down to 4 m/s over a distance of 40 m. What is the object's acceleration?
  15. A train traveling at 20 m/s needs to reach a velocity of 40 m/s to merge onto another track. If the train accelerates at a rate of 1 m/s², what distance will it cover during this acceleration?
  16. A particle with an initial velocity of 8 m/s accelerates for 6 s and covers a distance of 84 m. What is its acceleration?
  17. A truck accelerates from 10 m/s to 20 m/s over a period of 5 s. What is its displacement?
  18. An object starts from rest and covers a distance of 100 m in 5 s with constant acceleration. What is its final velocity?
  19. A car traveling at 28 m/s decelerates at a constant rate of -3.5 m/s². How far will the car travel before its velocity is 14 m/s?
  20. A remote-controlled airplane starts from rest and accelerates at 2.5 m/s². How long will it take for the airplane to travel 45 m?
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