---
title: "Momentum grade explained"
subject: "Transportation Engineering I"
chapter: 3
key: "semester_6/transportation_engineering_i:momentum-grade"
source: https://www.bctengineer.com/q/semester_6/transportation_engineering_i/momentum-grade
site: BCT Engineering
---

# Momentum grade explained

Transportation Engineering I · Chapter 3: Geometric Design of Highway

- Asked in: 2079 Ashwin, 4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/transportation_engineering_i/momentum-grade
- Whole chapter: https://www.bctengineer.com/md/semester_6/transportation_engineering_i/chapter-3

## Question

Explain momentum grade.

## Short answer

A momentum grade is a gradient steeper than the ruling gradient, used after a downgrade or level section where the vehicle gains speed to climb the short steep slope without significant speed loss. It reduces earthwork in rolling terrain, provided sight distance is adequate and the steep section is short with no stops.

## Answer

**Momentum grade** is a gradient steeper than the ruling gradient that can be used when it is preceded by a long downgrade or a level section, because the vehicle gains speed (kinetic energy) on the approach and uses that momentum to climb the steeper slope without losing much speed.

#### Conditions of use
- The approach must be a **downgrade or level section** long enough for vehicles to gain speed.
- The steep grade must be **short** (not long enough to stop heavy vehicles), and the vehicles must not be required to stop before it (no junction, no sharp curve).
- Used in rolling and hilly terrain to reduce the cost of earthwork.
- The sight distance should be adequate and the grade should not be used on a long upgrade.

#### Length of momentum grade
For a vehicle entering at speed $v_1$ and leaving at $v_2$ on a grade $G$ (as ratio):

$$
L = \frac{\gamma\,(v_1^2 - v_2^2)}{2g\,(G + f)}
$$

where $f$ is the rolling/friction resistance coefficient and $\gamma$ the coefficient of inertia of the vehicle's rotating parts.

```text
 \                 ____ steep grade (momentum)
  \  gain speed   /
   \_____________/ 
```

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is a momentum grade?
   A. A gradient steeper than the ruling gradient, used after a long downgrade or level section so vehicles use gained speed to climb it
   B. A gradient flatter than the ruling gradient, used to slow vehicles before a junction
   C. A gradient provided on a curve to compensate for curve resistance
   D. A gradient used only at the top of a summit curve to improve drainage
   Answer: A. Vehicles gain speed on the approach downgrade or level section and use that kinetic energy to climb the short steeper slope. (Key point: Definition of momentum grade)

2. Which condition must be satisfied before a momentum grade is used?
   A. It must follow a long upgrade so that engines are warmed up
   B. It must be long enough to make heavy vehicles stop and change gear
   C. The approach must be a long downgrade or level section and the vehicles must not have to stop before the grade
   D. It must be placed right after a sharp curve or a junction
   Answer: C. Speed must be gained on the approach, so the approach must be a downgrade or level and there must be no junction or sharp curve forcing a stop. (Key point: Conditions of use)

3. In L = gamma(v1^2 - v2^2)/(2g(G + f)), what does the term (v1^2 - v2^2) represent?
   A. The loss of kinetic energy (speed) between entering and leaving the steep grade
   B. The gain in potential energy of the road embankment
   C. The braking distance on the approach
   D. The friction force between tyre and road
   Answer: A. The vehicle enters at v1 and leaves at v2; the drop in v^2 is the kinetic energy used up on the steep grade, which sets the length. (Key point: Length of momentum grade)

4. Why are momentum grades used in rolling and hilly terrain?
   A. To increase vehicle speed on the upgrade
   B. To reduce the cost of earthwork by allowing a short steeper slope
   C. To provide a place for emergency stopping
   D. To remove the need for transition curves
   Answer: B. A short steeper grade avoids heavy cutting and filling, reducing earthwork cost. (Key point: Purpose of momentum grade)

## Related in Transportation Engineering I

- [Vertical curve 35 m headlight and momentum grade](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-80c-vc-headlight-35m) — Transportation Engineering I, chapter 3