# Kinematics: Describing Motion

Computer Science I · Physics · https://tryals.app/learn/computer-science-i/kinematics-describing-motion

## The Language of Motion

**Kinematics** describes _how_ things move, without asking _why_. Motion is a chain of derivatives and integrals linking three core quantities:

| Quantity | Symbol & Definition | Rate of Change |
| --- | --- | --- |
| **Position** | $x$: where an object is | — |
| **Velocity** | $v = \frac{dx}{dt}$: rate position changes | First derivative |
| **Acceleration** | $a = \frac{dv}{dt}$: rate velocity changes | Second derivative |

Speed is the magnitude of velocity. Integrating acceleration recovers velocity, and integrating velocity recovers position.

## Constant Acceleration & Pitfalls

For **constant acceleration** (like gravity $g \approx 9.8\,\text{m/s}^2$), calculus yields three exact **equations of motion**:

- $v = v_0 + a t$
- $x = x_0 + v_0 t + \tfrac{1}{2} a t^2$
- $v^2 = v_0^2 + 2a(x - x_0)$

Where $x_0$ and $v_0$ are initial position and velocity.

**Common pitfall:** Velocity and acceleration are **independent**. A tossed ball at its peak has $v = 0$ yet is still accelerating downward at $-g$. **Zero velocity does not mean zero acceleration.** Likewise, constant velocity means **zero** acceleration.

## Practice questions

3 of this lesson's 9 practice questions, with answers. The full set is in the app.

### 1. A game's physics behaves differently on a fast machine than a slow one, with objects tunnelling through walls at low frame rates. What is going wrong?

A. Slow machines have less accurate floating-point arithmetic
B. Objects accelerate faster when frames take longer, by conservation of momentum
C. The collision detection code runs at a different priority on slow machines
D. The timestep is tied to the frame rate, so a slow frame steps too far

**Answer:** D. The timestep is tied to the frame rate, so a slow frame steps too far

**Why:** Discrete-position collision checks miss anything that jumps the wall between frames, and a long frame means a long jump. Fixed timesteps and swept-volume tests are the standard fixes, a numerical-integration problem surfacing as a gameplay bug.

Page: https://tryals.app/practice/computer-science-i/kinematics-describing-motion/a-games-physics-behaves-differently-on-a-fast-machine-than-a-slow

### 2. A physics engine updates position from velocity and velocity from acceleration, each step multiplying by the timestep. Why does halving the timestep improve accuracy, and why does it not eliminate the error?

A. Because a shorter timestep gives the processor more time to compute each update accurately.
B. Because the error is caused by floating-point rounding, which halving the timestep halves.
C. Because the update assumes velocity is constant across the step.
D. Because the simulation becomes exact below a small enough timestep.

**Answer:** C. Because the update assumes velocity is constant across the step.

**Why:** Each update treats velocity as constant over the step, so it misses the change happening within it, and a shorter step improves the assumption while doubling the number of steps, so the improvement is a net effect. It never reaches zero, and below some timestep accumulated round-off grows faster than truncation shrinks. Which is why engines use Verlet or Runge–Kutta rather than plain Euler.

Page: https://tryals.app/practice/computer-science-i/kinematics-describing-motion/a-physics-engine-updates-position-from-velocity-and-velocity-from

### 3. A stone is thrown straight upward. Select every statement that is TRUE about its flight, ignoring air resistance.

A. Its acceleration reverses direction at the top
B. Its speed decreases on the way up and increases on the way down
C. Its acceleration is zero at the highest point
D. Its acceleration is downward throughout the flight
E. Its velocity is zero at the highest point

**Answer:** B. Its speed decreases on the way up and increases on the way down; D. Its acceleration is downward throughout the flight; E. Its velocity is zero at the highest point

**Why:** Gravity pulls downward at a constant g for the entire flight, so the acceleration never vanishes or reverses. Only the velocity changes sign, passing through zero at the top, which is precisely why zero velocity cannot imply zero acceleration.

Page: https://tryals.app/practice/computer-science-i/kinematics-describing-motion/a-stone-is-thrown-straight-upward-select-every-statement-that-is