Start where you are, then drift by v every second: Picture a drone hovering at a starting point, then gliding off at a steady velocity.
Its position after t seconds is
r(t) = r₀ + t·v
— the starting position r₀ plus the velocity vector v added on t times. Each second, the drone's coordinates shift by exactly the components of v. To find where it is at any time, just substitute the value of t.
IB-style question — find a position from r(t)
A delivery drone starts at the point (2, 1) km and flies with constant velocity v = (3, 4) km/h, where t is in hours.
Write down its position vector r(t), then find where it is after 2 hours.
Step by step
- Position = start + t × velocity.
- Substitute t = 2.
Final answer
r(t) = (2 + 3t, 1 + 4t); after 2 hours the drone is at (8, 9) km.
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Velocity is a vector; speed is its length: Velocity v is a vector — it carries both direction and rate. If the position is written as r(t) = r₀ + t·v, the velocity is simply the vector multiplying t (the coefficient of t in each coordinate).
Speed is a single number — how fast, ignoring direction. It is the magnitude (length) of the velocity vector:
speed = |v| = √(vₓ² + vy²) (Pythagoras on the components).
IB-style question — read off velocity, find speed
An ice-skater's position (in metres) is r(t) = (4 + 6t, 12 − 8t), with t in seconds.
Find the skater's velocity vector and speed.
Step by step
- Velocity is the vector multiplying t (the coefficient of t in each coordinate).
- Speed = length of v, by Pythagoras on the components.
- Evaluate.
Final answer
Velocity v = (6, −8) m/s; speed = 10 m/s. (The skater moves right and downward at 10 m/s.)
IB-style question — integrate acceleration
A ball is launched from the origin with acceleration a(t) = (2, −10) m s⁻² and initial velocity v(0) = (8, 15) m s⁻¹.
Find its position vector r(t), and the horizontal distance travelled when it returns to ground level (y = 0).
Step by step
- Integrate acceleration for velocity; the constant of integration is v(0).
- Integrate again for position; the constant is r(0) = (0, 0).
- Ground level: set the y-component to 0 and solve.
- Horizontal distance is the x-component at t = 3.
Final answer
r(t) = (t² + 8t, −5t² + 15t); it lands after 3 s, 33 m horizontally from the start. (Integrate twice; initial velocity and position are the constants.)
Velocity vs speed — don't mix them up: Velocity = a vector like (6, −8) (it has components and a direction).
Speed = one number like 10 (always positive, no direction).
If a question asks 'how fast', it wants the speed (a magnitude); if it asks for 'velocity', leave it as a vector.