Equations of Uniformly Accelerated Motion

Equations of Uniformly Accelerated Motion

Subtopics

• First Equation of Motion
• Second Equation of Motion
• Third Equation of Motion
• Free-Fall Motion

1. Learning Outcomes

After studying this topic, students should be able to:

• Define uniformly accelerated motion.
• Identify initial velocity, final velocity, acceleration, time, and displacement.
• State and derive the three equations of motion.
• Select the correct equation for a given numerical problem.
• Use an appropriate sign convention.
• Apply the equations to accelerating and decelerating objects.
• Explain free fall and acceleration due to gravity.
• Solve problems involving dropped and vertically projected objects.

2. Definition

3. Conditions for Using the Equations

The three equations of motion can be applied when:

• Acceleration is constant.
• Motion occurs along a straight line.
• All quantities are measured in a consistent unit system.
• A clear positive direction has been selected.

They should not be applied directly when acceleration continuously changes.

4. Key Terms

Initial Velocity

The velocity at the beginning of the selected time interval. u = initial velocity

Final Velocity

The velocity at the end of the selected time interval. v = final velocity

Acceleration

The rate of change of velocity. a = v − ut

Time

The duration for which the motion is studied. t = time interval

Displacement

The change in the object’s position in a particular direction. s = displacement

Deceleration

Acceleration acting opposite to the selected positive direction. It is represented by a negative value of a.

5. Symbols and SI Units

Symbols and SI Units

6. Main Equations of Motion

First Equation

v = u + at

Second Equation

s = ut + ½at²

Third Equation

v² = u² + 2as

Average-Velocity Relation

s = (u + v2) t

Uniformly Accelerated Motion

18. Formula-Selection Guide

Formula-Selection Guide

Quick Method

• Write all known quantities.
• Identify the required quantity.
• Identify the quantity that is completely absent.
• Select the equation that does not contain the absent quantity.

28. Important Facts

• The equations apply only to constant acceleration.
• Deceleration is represented by negative acceleration.
• Displacement and distance are not always identical.
• A negative velocity indicates direction.
• Gravity always acts downward.
• At maximum height, vertical velocity is zero.
• Acceleration at maximum height is still g downward.
• In ideal free fall, objects of different masses have the same acceleration.
• An object returning to its launch level has the same speed as its initial speed when air resistance is ignored.
• Its final velocity has the opposite direction to its initial velocity.

29. Key Logic

The equations of motion do not change during free fall. Only the symbols and signs are adjusted: a → g or: a → −g The sign depends on the selected positive direction. Never memorize gravity as always negative. Gravity is negative only when upward has been selected as positive.

30. Real-Life Applications

These equations are used to study:

• Accelerating vehicles
• Braking and stopping distance
• Objects dropped from buildings
• Vertically thrown balls
• Lift and elevator motion
• Runway motion of aircraft
• Motion of trains
• Laboratory trolley experiments
• Objects moving on straight inclined paths
• Short vertical motion near Earth’s surface

31. Tips and Tricks

• Write u, v, a, t, s before choosing an equation.
• Convert kilometres per hour into metres per second when required.
• Select the positive direction before inserting signs.
• Use u = 0 only when an object actually starts from rest.
• Use v = 0 at the highest point of vertical motion.
• Use negative acceleration for slowing motion only when acceleration opposes the positive direction.
• Never mix upward-positive and downward-positive equations in one calculation.
• Use the positive root for elapsed time unless the negative root has a relevant mathematical meaning.
• Include both magnitude and direction in velocity answers.

32. Common Mistakes

• Applying the equations when acceleration is not constant.
• Confusing initial and final velocity.
• Writing g = 0 at the highest point.
• Treating displacement as always positive.
• Using a positive acceleration for braking without considering direction.
• Forgetting the square on t in the second equation.
• Forgetting the square on u and v in the third equation.
• Mixing g = 9.8 m s⁻² and g = 10 m s⁻² in one problem.
• Writing the velocity unit as m s⁻².
• Giving two positive times for an upward-and-downward journey without interpreting them.

33. Remember Box

34. Exam Facts

• The first equation does not contain displacement.
• The second equation does not contain final velocity.
• The third equation does not contain time.
• The area under a velocity–time graph represents displacement.
• The slope of a velocity–time graph represents acceleration.
• At maximum height, v = 0, but a = −g when upward is positive.
• A dropped object has u = 0.
• Negative acceleration does not always mean that an object is slowing down; directions must be compared.
• Use one consistent sign convention throughout a solution.

35. Short-Question Preparation

Short-Question Preparation

36. Long-Question Preparation

Derive the three equations of uniformly accelerated motion.

Consider an object moving with constant acceleration a. Let its initial velocity be u. Let its final velocity after time t be v. Acceleration is the rate of change of velocity: a = v − ut Rearranging gives: v = u + at This is the first equation of motion. For constant acceleration, average velocity is: v_avg = u + v2 Displacement is average velocity multiplied by time: s = (u + v2) t Substituting v = u + at: s = (u + (u + at)) / 2 × t Simplifying gives: s = ut + ½at² This is the second equation. From the first equation: t = v − ua Substitute this value into the average-velocity relation: s = (u + v2) × (v − ua) Therefore: 2as = (u + v)(v − u) Using the difference of squares: 2as = v² − u² Rearranging gives: v² = u² + 2as This is the third equation of motion. All three equations require constant acceleration.

40. Sticky Notes

Sticky Notes

• Constant acceleration is required.
• First equation excludes s.
• Second equation excludes v.
• Third equation excludes t.
• Dropped from rest means u = 0.
• At maximum height, v = 0.
• Gravity remains downward throughout the motion.
• Choose a sign convention before substitution.
• Use one calculation step per line.
• Always write the final unit and direction.

41. Quick Revision

v = u + at s = ut + ½at² v² = u² + 2as s = (u + v2) t For an object dropped from rest: v = gt h = ½gt² v² = 2gh

42. 60-Second Revision

• List the known and required quantities.
• Select a positive direction.
• Insert the correct signs.
• Choose the equation that excludes the unwanted quantity.
• For upward-positive free fall, use a = −g.
• For downward-positive falling motion, use a = +g.
• At maximum height, use v = 0, not g = 0.
• Check units and direction in the final answer.

43. Topic Summary

The equations of uniformly accelerated motion describe straight-line motion with constant acceleration. The first equation relates velocity and time, the second relates displacement and time, and the third relates velocity and displacement without using time. The same equations describe free fall when acceleration is replaced by +g or −g according to the chosen sign convention.

44. Suggested Ask-the-Tutor Questions

• How do I select the correct equation of motion?
• Why is acceleration negative during braking?
• Why is gravity not zero at maximum height?
• How do I choose a sign convention for free fall?
• What is the difference between a dropped and projected object?
• How are the equations derived from a velocity–time graph?

45. References

• Prescribed Class 11 Physics textbook
• Official Class 11 Physics curriculum
• School physics laboratory manual
• Teacher-provided classroom notes