Straight line or curve? Conditions for curved motion
A body moves in a straight line if there is no net force, or if the net force is exactly along its velocity (or opposite to it). Then only the speed changes.
A body moves on a curve (curvilinear motion) when the net force is not along the velocity. The sideways part of the force turns the velocity. At every point the velocity points along the tangent to the path, and the path bends toward the side the force pushes.
Two examples: a ball thrown at an angle (force of gravity is down, velocity is slanted) and a stone whirled on a string (force toward the centre, velocity along the tangent).
Velocity is always along the tangent
If you cut the string of a whirling stone, it flies off along the line it was moving in at that moment. That line is the tangent. This shows that the velocity at any point is along the tangent. In curved motion the velocity changes direction, so the body is accelerating even if its speed stays the same.
Composition of motion
When a body takes part in two motions at once, its real motion is the composition (vector sum) of the two. The key rule is independence: each motion goes on as if the other did not exist. Displacements, velocities and accelerations add as vectors.
Boat in a river: the boat's speed in still water is u (across) and the river's speed is w (along the bank). The actual speed is v = √(u² + w²), and the angle with the bank satisfies tan θ = u/w.
The time to cross a river of width D is t = D/u. The river's flow w only moves the boat downstream by w × t. It does not change the time.
Resolution of motion
Resolution is the reverse: split one motion into two at right angles. A velocity v at angle θ to the horizontal has components vx = v cos θ and vy = v sin θ. The two components can be studied separately and then put back together. This is how we later handle projectiles: horizontal motion at constant speed and vertical motion with gravity.
Special cases for a boat: to cross in the shortest time, head straight across (t = D/u). To land directly opposite you must aim upstream at an angle θ from the perpendicular with sin θ = w/u; this needs u > w.
Try it at home
Roll a marble straight across a tilted table or a sheet of paper while a friend slowly pulls the paper sideways. Mark the marble's path with a pencil dot every second. It is a slanted line. Pull faster and the line leans more, yet the marble reaches the far edge at the same time.
Key formulas and definitions
- Net force along velocity → straight path
- Net force not along velocity → curved path
- Velocity is along the tangent to the path
- v = √(u² + w²), tan θ = u/w (boat across a river)
- Crossing time t = D/u; drift = w × t
- Components: vₓ = v cos θ, v_y = v sin θ
- Shortest path across: sin θ = w/u (aim upstream, u > w)
Worked examples
1. A boat heads straight across a river at 4 m/s in still water. The river flows at 3 m/s. Find the boat's actual speed and its angle with the bank.
v = √(4² + 3²) = 5 m/s. tan θ = u/w = 4/3, so θ ≈ 53° with the bank.
2. The river is 100 m wide. With u = 4 m/s across and w = 3 m/s, find the crossing time and the drift.
t = D/u = 100/4 = 25 s. Drift = w × t = 3 × 25 = 75 m downstream.
3. A ball is kicked at 10 m/s at 30° above the ground. Find its horizontal and vertical velocity components.
vₓ = 10 cos 30° = 8.66 m/s and v_y = 10 sin 30° = 5 m/s.
4. A plane heads north at 80 m/s in air. A wind blows towards the east at 60 m/s. What is its speed over the ground?
v = √(80² + 60²) = √10000 = 100 m/s, at an angle east of north with tan = 60/80.
5. A boat with u = 5 m/s wants to land directly opposite on a river with w = 3 m/s and width 120 m. In which direction must it head, and how long does it take?
sin θ = w/u = 3/5, so θ ≈ 37° upstream from the perpendicular. Speed across = √(5² − 3²) = 4 m/s. t = 120/4 = 30 s.
6. A ball rolls off a table 0.8 m high with a horizontal speed of 2 m/s. Find the time to fall, the horizontal distance and the speed when it lands (g = 10 m/s²).
Vertical: 0.8 = ½ × 10 × t², so t = 0.4 s. Horizontal distance = 2 × 0.4 = 0.8 m. Vertical speed = 10 × 0.4 = 4 m/s. Speed = √(2² + 4²) = √20 ≈ 4.5 m/s.
Common mistakes
- Thinking a body needs a force along its path to move on a curve. The force must be off the line of the velocity.
- Thinking a faster river makes the boat take longer to cross. Crossing time is D/u only.
- Adding speeds with plain addition when the motions are at right angles. Use v = √(u² + w²).
- Drawing the velocity of a circling body toward the centre. It is along the tangent; the force and acceleration point to the centre.