Gerak Satu Dimensi-Full

One-Dimensional Motion

One-dimensional motion is motion in which an object’s position changes along a single axis. It can occur horizontally or vertically. An object is considered to be moving when its position changes relative to a chosen reference point.

Distance and Displacement

• Distance is the total length of the path traveled. It is a scalar, meaning it has magnitude but no direction.
• Displacement is the change from an object’s initial position to its final position. It is a vector, meaning it has both magnitude and direction.

In one dimension:

[ \Delta x = x_f - x_i ]

where (x_i) and (x_f) are the initial and final positions.

Example: A person walks 2 m right, 4 m left, and then 3 m right. The total distance is (2 + 4 + 3 = 9) m. The final position is 1 m to the right of the starting point, so the displacement is 1 m right.

Train example: A train travels from 3 km east of a bridge to 26 km west of it. Taking the bridge as the origin, its displacement is (-26 - 3 = -29) km, or 29 km west. Its distance traveled is 29 km.

Speed and Velocity

• Average speed is total distance divided by total elapsed time. It is a scalar and is nonnegative.

[ \text{average speed} = \frac{\text{total distance}}{\Delta t} ]

• Average velocity is displacement divided by elapsed time. It is a vector and includes direction.

[ \bar v = \frac{\Delta x}{\Delta t} ]

On a position–time graph, the slope represents average velocity over an interval. The magnitude of that velocity is the average speed when motion is in one direction throughout the interval.

Example: Someone runs 200 m east at 5 m/s, then 280 m west at 4 m/s.

• Total distance: 480 m
• Displacement: 80 m west
• Total time: (200/5 + 280/4 = 110) s
• Average speed: (480/110 \approx 4.36) m/s
• Average velocity: (80/110 \approx 0.73) m/s west

Instantaneous Velocity and Acceleration

Instantaneous velocity is an object’s velocity at a particular moment. On a position–time graph, it is the slope of the tangent to the curve at that moment.

[ v = \frac{dx}{dt} ]

Acceleration describes how velocity changes over time. It is a vector, so its sign and direction matter.

Average acceleration is:

[ \bar a = \frac{\Delta v}{\Delta t} ]

Instantaneous acceleration is:

[ a = \frac{dv}{dt} = \frac{d^2x}{dt^2} ]

On a velocity–time graph, acceleration is the slope; instantaneous acceleration is the slope of the tangent. Acceleration in the direction of motion makes an object speed up, while acceleration opposite the motion makes it slow down.

Example: For a car with position (x(t) = 2.4t^2 - 0.12t^3), differentiating the position function gives its velocity. The car’s average velocity from (t = 0) to (10) s is 12 m/s, and its instantaneous velocity at (t = 5) s is 15 m/s. The car is at rest again at about (t = 13.3) s.

Uniform Straight-Line Motion

Uniform straight-line motion is motion along a straight path at constant velocity. An object’s position can be found using:

[ x_f = x_i + v\Delta t ]

On a position–time graph, uniform straight-line motion appears as a straight line whose slope is the velocity.

Example: A runner covers 30 m in 5 s at constant speed:

[ v = \frac{30}{5} = 6\text{ m/s} ]

Uniformly Accelerated Straight-Line Motion

In uniformly accelerated straight-line motion, acceleration is constant. The lesson distinguishes between:

• Accelerated motion, where speed increases.
• Decelerated motion, where speed decreases. In the examples, deceleration is represented by negative acceleration.

The equations are:

[ v_f = v_i + at ]

[ x_f = x_i + v_it + \frac{1}{2}at^2 ]

[ v_f^2 = v_i^2 + 2a(x_f - x_i) ]

Example: A car starts from rest and reaches 20 m/s after traveling 120 m with constant acceleration. Using the third equation gives (a \approx 1.67\text{ m/s}^2). Using (v_f = v_i + at) gives a travel time of about 12 s.

Free Fall

Free fall is vertical motion under the influence of gravity, with air resistance ignored. In a vacuum, objects of different masses and shapes fall together. Their acceleration does not depend on mass or shape; it is the local gravitational acceleration (g). The same constant-acceleration equations apply, with vertical position and (g) used in place of horizontal position and acceleration.

Example: A brick dropped from rest falls for 2.5 s, using (g = 9.8\text{ m/s}^2):

• Height fallen: (\frac{1}{2}gt^2 \approx 30.6) m
• Speed just before impact: (gt = 24.5) m/s downward

Finding Position and Velocity by Integration

When acceleration is given as a function of time, integration can be used to find velocity and then position:

[ v(t) = v_0 + \int_0^t a(t)\,dt ]

[ x(t) = x_0 + \int_0^t v(t)\,dt ]

For the example’s acceleration function, the lesson derives velocity and position functions, then finds maximum speed by setting (a = dv/dt = 0). The maximum speed is about 39.1 m/s.

Speakers and Sources Featured

• Mrs. Deltawati — the main lecturer and narrator.
• Unidentified voices or presenters in inserted illustrative clips — used for demonstrations and explanations, including the free-fall comparison. They are not identified by name in the subtitles.