The Five Equations of Motion: Why Your Textbook Might Be Missing One

Introduction

When learning about motion with constant acceleration (like a car speeding up or a ball rolling down a hill), we use special equations to connect five key quantities:

• u = starting speed (initial velocity)
• v = final speed
• a = acceleration
• s = distance traveled (displacement)
• t = time taken

Since there are five variables, we might expect five equations—one for each case where we don’t use one of the variables. Most textbooks give three or four equations, but there’s actually a fifth one that’s often left out! Let’s explore why.

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The Three (or Four) Common Equations

Most textbooks teach these three main equations first:

1. v = u + at (connects speed, acceleration, and time)
2. s = ut + ½ at² (connects distance, starting speed, acceleration, and time)
3. v² = u² + 2as (connects speed, acceleration, and distance)

Some books add a fourth equation:

4. s = ½(u + v)t (connects distance, starting speed, final speed, and time)

But there’s a fifth equation that’s rarely mentioned:

5. s = vt − ½ at²

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Why Is the Fifth Equation (s = vt − ½ at²) Missing?

1. It’s Similar to Another Equation
   • The second equation (s = ut + ½ at²) already relates distance (s) to initial speed (u), acceleration (a), and time (t).
   • The fifth equation (s = vt − ½ at²) is just a rearranged version where we use final speed (v) instead of initial speed (u).
2. We Can Derive It from the First Equation
   • If we know v = u + at, we can rewrite u = v − at.
   • Plugging this into s = ut + ½ at² gives:
     s = (v − at)t + ½ at² = vt − at² + ½ at² = vt − ½ at²
   • Since we can get it from other equations, some books skip it.
3. It’s Less Commonly Needed
   • Most problems give initial speed (u) rather than final speed (v), so s = ut + ½ at² is more useful.
   • The fifth equation is handy only in special cases where we know v but not u.

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Should You Learn All Five?

Yes! Even though the fifth equation isn’t always in textbooks, it can save time in some problems. For example:

*A car slows down to 10 m/s (v) after braking for 4 seconds (t) with an acceleration of −2 m/s² (a). How far did it travel?*

Using s = vt − ½ at²:

• s = (10)(4) − ½ (−2)(4)² = 40 + 16 = 56 meters

Without this equation, you’d have to first find u using v = u + at, which takes an extra step.

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Conclusion

Textbooks usually focus on the three or four most useful equations. The fifth one (s = vt − ½ at²) is often left out because:

• It’s very similar to s = ut + ½ at², just using v instead of u.
• You can derive it from other equations.
• It’s not needed as often.

But knowing all five makes you a smarter problem-solver!