Ever wondered why a ball arcs beautifully when you throw it, while a car just zooms straight ahead?
💡 In Simple Words: Motion in a straight line means moving along one direction, like a train on tracks. Motion in a plane means moving in two directions at once, like a soccer ball kicked into the air. Both follow simple rules that let us predict where they’ll be.
Motion in a Straight Line (1‑D Kinematics)
Key ideas you’ll hear a lot
Displacement is the shortest straight‑line distance from start to finish, with a direction. Think of it as the “as‑the‑crow‑flies” distance.
Distance is how much ground you actually covered, ignoring direction. It’s like the miles on a road trip.
Speed tells how fast you go, ignoring direction. It’s distance divided by time.
Velocity adds direction to speed. It’s displacement divided by time.
Acceleration measures how quickly velocity changes. If you press the gas, you feel the car speed up—that’s positive acceleration.
Equations for uniform (constant) acceleration
When acceleration stays the same, these four equations let you solve any problem without calculus:
- v = u + a t
- s = u t + ½ a t²
- v² = u² + 2 a s
- s = ½ (u + v) t
Here, u is initial velocity, v final velocity, a acceleration, t time, and s displacement.
Worked example: A car accelerates
A car starts from rest (u = 0) and accelerates uniformly at 2 m/s² for 5 seconds. Find its final speed and the distance covered.
Using v = u + a t:
v = 0 + (2 m/s²)(5 s) = 10 m/s.
Now use s = u t + ½ a t²:
s = 0 + ½ (2)(5²) = 0.5 * 2 * 25 = 25 m.
So after 5 seconds the car is moving at 10 m/s and has travelled 25 m.
Motion in a Plane (2‑D Kinematics)
Why two directions matter
When an object moves in two dimensions, we treat the motion as two independent one‑dimensional motions happening at the same time—one horizontal, one vertical.
Think of a skateboarder doing a jump: the forward glide is horizontal, while the lift‑off and landing are vertical.
Projectile motion basics
A projectile is any object that moves under the influence of gravity alone after leaving the launch point. Gravity pulls straight down with an acceleration of 9.8 m/s² (≈10 m/s² for quick calculations).
Key points:
- The horizontal motion has zero acceleration (ignoring air resistance).
- The vertical motion has constant downward acceleration (g).
- Both components share the same time of flight.
Equations for the horizontal (x) direction
Since aₓ = 0, the equations simplify to:
- vₓ = uₓ (velocity stays constant)
- x = uₓ t (distance = speed × time)
Equations for the vertical (y) direction
Here a_y = –g (negative because it points down). Use the same four equations as before, but replace a with –g and s with y (vertical displacement).
Worked example: Throwing a ball
You throw a ball with an initial speed of 20 m/s at a 30° angle above the horizontal. Find its maximum height and the horizontal range (how far it lands).
First split the speed into components:
uₓ = 20 cos30° ≈ 20 × 0.866 = 17.3 m/s
u_y = 20 sin30° = 20 × 0.5 = 10 m/s
Maximum height occurs when vertical velocity becomes zero:
v_y² = u_y² – 2 g h → 0 = 10² – 2(10)h → h = 5 m.
Time to reach the top: t_up = u_y / g = 10 / 10 = 1 s.
Total time of flight is twice that (symmetry): t_total = 2 s.
Horizontal range: x = uₓ t_total = 17.3 × 2 ≈ 34.6 m.
So the ball rises to 5 m and lands about 35 m away.
Quick Comparison: 1‑D vs 2‑D Motion
| Feature | 1‑D Motion | 2‑D Motion |
|---|---|---|
| Direction | Single line (straight) | Two perpendicular lines (horizontal & vertical) |
| Acceleration | Can be constant or zero | Horizontal a = 0, vertical a = –g (gravity) |
| Key equations | Four kinematic formulas | Separate horizontal & vertical equations; combine using time |
| Typical examples | Car accelerating, falling object dropped | Projectile motion, inclined plane motion |
Bullet‑point Summary
- Displacement has direction; distance does not.
- Speed = distance/time; velocity = displacement/time.
- Uniform acceleration lets you use four simple equations.
- In 2‑D motion, treat horizontal and vertical parts independently.
- Gravity provides a constant downward acceleration of ~10 m/s².
- Horizontal velocity stays constant; vertical velocity changes linearly.
- Time of flight is the same for both components, linking them together.
📝 Likely Exam Questions
- Define displacement and differentiate it from distance.
Answer: Displacement is the straight‑line change in position with a direction, while distance is the total path length travelled, ignoring direction. - A stone is dropped from a 45 m high tower. Find the time it takes to hit the ground.
Answer: Use s = ½ g t² → 45 = 0.5 × 10 × t² → t² = 9 → t = 3 s. - Calculate the range of a projectile launched at 25 m/s at 40°.
Answer: uₓ = 25 cos40° ≈ 19.1 m/s, u_y = 25 sin40° ≈ 16.1 m/s. Time of flight = 2u_y/g ≈ 2×16.1/10 = 3.22 s. Range = uₓ × t ≈ 19.1 × 3.22 ≈ 61.5 m. - Using v² = u² + 2as, find the final speed of a car that starts from rest and accelerates at 3 m/s² over 50 m.
Answer: v² = 0 + 2×3×50 = 300 → v ≈ 17.3 m/s. - Explain why the horizontal component of velocity remains constant in projectile motion.
Answer: In the absence of air resistance, no horizontal force acts on the projectile, so according to Newton’s first law, its horizontal velocity does not change.