Motion in two dimensions is described by a position vector that changes with time. Calculus turns that changing position into velocity and acceleration vectors, which tell us how fast the particle moves and how its motion is changing. These vectors are essential for understanding projectiles, circular motion, orbital motion, and any path that curves through space.
A velocity vector always points tangent to the path, while acceleration can point in a different direction.
If the position is r(t) = <x(t), y(t)>, then velocity is the first derivative v(t) = r'(t) and acceleration is the second derivative a(t) = r''(t). The speed is the magnitude of velocity, so it measures how quickly distance along the path is increasing. Acceleration describes changes in speed, direction, or both, which is why it may point inward on a curve even when the object moves forward.
By separating motion into vector components, calculus lets us analyze complex curved motion using familiar one-dimensional derivatives.
Understanding Calculus: Velocity and Acceleration Vectors
A vector function can be treated as two linked one dimensional functions. One function tracks horizontal position and the other tracks vertical position. Differentiate each component separately at the same time value.
This works because the horizontal and vertical directions are independent reference directions. A particle may move right while moving down, for example. Its velocity records both changes together.
The units help keep the ideas clear. If position is measured in metres and time in seconds, velocity is measured in metres per second.
Acceleration is measured in metres per second per second. Checking units often catches a mistaken derivative or an incorrect interpretation of a graph.
Velocity describes an instant, not a whole trip. Over a time interval, displacement divided by elapsed time gives average velocity. As the interval becomes very small, that average approaches instantaneous velocity.
This limiting process is the reason derivatives are useful for motion. A graph of horizontal position against time can have a positive slope while the graph of vertical position has a negative slope.
The resulting velocity points down and to the right. Reading component graphs is a practical skill because many motion sensors, video analysis tools, and computer simulations store data as separate coordinates.
Acceleration deserves careful attention because it does not simply mean an object is speeding up. Compare the acceleration direction with the velocity direction. If acceleration has a component in the same direction as velocity, speed increases.
If it has a component opposite velocity, speed decreases. If acceleration is perpendicular to velocity, it changes direction without changing speed at that instant. A car travelling around a level bend is a familiar case.
Even with a steady speedometer reading, the car has acceleration toward the centre of the bend. Friction between the tyres and road provides much of the inward force needed for that turn.
For curved paths, it is useful to separate acceleration into a tangential part and a normal part. The tangential part lies along the path and controls the change in speed. The normal part points toward the inside of the curve and controls the turning of the velocity vector.
A thrown ball shows both effects. Gravity gives it a constant downward acceleration, while its velocity changes from upward to downward. At the highest point, the vertical velocity component is zero, but the acceleration is still downward.
Zero velocity at one instant does not automatically mean zero acceleration. When solving problems, find the vector quantities first, then calculate magnitudes or directions only when the question requires them.
Key Facts
- Position vector in 2D: r(t) = <x(t), y(t)>
- Velocity vector: v(t) = r'(t) = <dx/dt, dy/dt>
- Acceleration vector: a(t) = v'(t) = r''(t) = <d2x/dt2, d2y/dt2>
- Speed is the magnitude of velocity: speed = |v(t)| = sqrt((dx/dt)^2 + (dy/dt)^2)
- The velocity vector is tangent to the position curve when the motion is smooth and |v(t)| is not zero.
- Acceleration can change speed, direction, or both, and it often has a component pointing toward the inside of a curved path.
Vocabulary
- Position vector
- A vector r(t) that gives the location of a moving object relative to an origin at time t.
- Velocity vector
- The derivative of the position vector, giving the object's instantaneous direction of motion and rate of change of position.
- Acceleration vector
- The derivative of the velocity vector, giving how the object's velocity changes with time.
- Speed
- The scalar magnitude of the velocity vector, measured as distance traveled per unit time.
- Tangent vector
- A vector that points along the direction of a curve at a specific point.
Common Mistakes to Avoid
- Treating velocity and speed as the same thing is wrong because velocity has direction while speed is only a magnitude.
- Drawing the velocity vector toward the center of the curve is wrong because velocity points tangent to the path, not inward, for smooth motion.
- Assuming acceleration must point in the direction of motion is wrong because acceleration points in the direction that velocity is changing.
- Forgetting to differentiate each component separately is wrong because r(t) = <x(t), y(t)> requires dx/dt and dy/dt for velocity, then second derivatives for acceleration.
Practice Questions
- 1 A particle has position r(t) = <3t^2, 4t>. Find v(t), a(t), and the speed at t = 2.
- 2 A particle moves with r(t) = <cos t, sin t>. Find v(t), a(t), and the speed at t = pi/2.
- 3 A car moves around a curve at constant speed. Explain why its acceleration is not zero even though its speed does not change.