Motion in a Straight Line Short Notes for Engineering Entrance Exams

MOTION IN A STRAIGHT LINE

---

1. Basic Terms

Position (x)

Location of a particle on a line.

Displacement (Δx)

Change in position.
Vector quantity.

$$\mathrm{\Delta }x=x_2-x_{\mathrm{1}}$$

Distance

Total path length (scalar).

---

2. Speed & Velocity

Average Speed

$$\text{Avg Speed}=\frac{\text{Total Distance}}{\text{Total Time}}$$

Average Velocity

$$\text{Avg Velocity}=\frac{\text{Displacement}}{\text{Total Time}}$$

Instantaneous Velocity

$$v=\frac{dx}{dt}$$

---

3. Acceleration

Average Acceleration

$$a=\frac{\mathrm{\Delta }v}{\mathrm{\Delta }\mathrm{t}}$$

Instantaneous Acceleration

$$a=\frac{dv}{d\mathrm{t}}$$

If velocity decreases → negative acceleration (retardation).

---

4. Equations of Motion (For Constant Acceleration)

$$v = u + at$$
$$s = ut + \frac{1}{2}at^2$$
$$v^2 = u^2 + 2as$$
$$s=\frac{v+u}{2}t$$

Where:
u = initial velocity
v = final velocity
a = acceleration
s = displacement
t = time

---

5. Graphs (Very Important for Exams)

(A) x–t graph

• Slope = velocity

• Straight line → uniform velocity

• Curve → variable velocity

(B) v–t graph

• Slope = acceleration

• Area under graph = displacement

• Straight line with slope → uniform acceleration

(C) a–t graph

• Area = change in velocity

---

6. Free Fall (a = g = 9.8 m/s² downward)

Taking downward positive:

$$v = u + gt$$
$$h = ut + \frac{1}{2}gt^2$$
$$v^2=u^2+2gh$$

If thrown upward, use a = –g.

---

7. Relative Motion in 1D

If two bodies A and B move on same line:

Relative Velocity of A w.r.t B

$$v_{AB}=v_A-v_{\mathrm{B}}$$

Key Cases

• Moving in same direction → relative speed = |vA − vB|

• Moving in opposite direction → relative speed = vA + vB

Used in overtaking and meeting-time problems.

---

8. Average Speed in Special Cases

Case: Two equal distances with speeds v₁ and v₂

$$\text{Avg Speed} = \frac{2v_1v_2}{v_1 + v_2}$$

(Harmonic mean — frequently asked)

---

9. Important Concepts

Uniform Motion

Velocity constant → a = 0.

Uniformly Accelerated Motion

Acceleration constant → use equations of motion.

Instantaneous Rest

Velocity v = 0 at that moment only — NOT start or end of motion necessarily.

Projectile in Vertical Direction

• Upward motion → retardation (a = –g)

• At max height, v = 0

---

10. Most Expected Exam Question Types

1. Numerical using equations of motion

2. Distance-time and velocity-time graphs

3. Relative velocity problems

4. Average speed for multiple segments

5. Free fall / upward throw numericals

6. Time to overtake, meeting point