# Kinematics in One Dimension

Physics I · Principles of Mechanics · https://tryals.app/learn/physics-i/kinematics-in-one-dimension

## Kinematics & Rates

Close your eyes on a smoothly cruising train and you feel nothing, but the slightest braking registers instantly. That asymmetry is **kinematics**: describing motion without asking what causes it.

Motion in a straight line uses three core quantities. **Position** ($x$) is where you are, **velocity** ($v$) is how position changes, and **acceleration** ($a$) is how velocity changes:

$$v = \frac{dx}{dt}, \qquad a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$$

Each layer is the rate of change of the one before it.

## Constant Acceleration & Graphs

When acceleration is constant, use the **SUVAT** equations:

| Equation | Missing Variable |
| --- | --- |
| $v = v_0 + at$ | position ($x$) |
| $x = x_0 + v_0 t + \frac{1}{2}at^2$ | final velocity ($v$) |
| $v^2 = v_0^2 + 2a(x-x_0)$ | time ($t$) |
| $x = x_0 + \frac{1}{2}(v_0+v)t$ | acceleration ($a$) |

**Graphical analysis** connects geometry to motion:
- **Slope** of $x$-$t$ graph = velocity.
- **Slope** of $v$-$t$ graph = acceleration.
- **Area** under $v$-$t$ graph = displacement.

**Free fall** is motion under gravity alone, where $a = -g = -9.8\text{ m/s}^2$ (upward positive).

> **Common pitfall:** Negative acceleration does not mean slowing down. A falling ball has $a < 0$ while speeding up. Slowing down means $v$ and $a$ have **opposite** signs.

## Practice questions

3 of this lesson's 12 practice questions, with answers. The full set is in the app.

### 1. A car speeding up and a car slowing down can both have positive acceleration. How?

A. Because the sign is the direction of the change, not the motion
B. Because deceleration is not a physical quantity, only a description
C. Because the sign depends on the observer rather than on the motion
D. Because acceleration is always positive when the speed changes

**Answer:** A. Because the sign is the direction of the change, not the motion

**Why:** Speeding up means $v$ and $a$ share a sign; slowing means they differ. "Deceleration" names the relationship, not a sign, which is why it is avoided in careful work.

Page: https://tryals.app/practice/physics-i/kinematics-in-one-dimension/a-car-speeding-up-and-a-car-slowing-down-can-both-have-positive

### 2. Sort into Scalar or Vector quantities.

**Answer:**

- Scalar: Speed, Distance, Time
- Vector: Velocity, Displacement, Acceleration

**Why:** Vectors have both magnitude and direction; scalars have only magnitude.

Page: https://tryals.app/practice/physics-i/kinematics-in-one-dimension/sort-into-scalar-or-vector-quantities

### 3. A ball thrown straight up has zero velocity at the top but acceleration $-g$ throughout. Why does the acceleration not vanish where the motion does?

A. Because acceleration is defined as an average and cannot be zero at an instant
B. Because the ball is still moving horizontally at the top
C. Because air resistance sustains the acceleration when gravity is balanced
D. Because acceleration is set by the forces, not by the velocity

**Answer:** D. Because acceleration is set by the forces, not by the velocity

**Why:** If $a$ were zero at the top the ball would stay there. Acceleration is fixed by the net force, and that force is $-mg$ all the way up and all the way down.

Page: https://tryals.app/practice/physics-i/kinematics-in-one-dimension/a-ball-thrown-straight-up-has-zero-velocity-at-the-top-but