Speed, motion graphs Notes

How Do Motion Graphs Help Us Recognise Patterns In Movement?

1. When you watch a juggler, you cannot track every ball, but your brain still predicts where the balls will be.
2. Physics turns that intuition into measurable quantities and graphs so that motion can be described precisely and predictions can be tested.
3. Motion occurs when an object changes its position over time relative to a chosen reference point.

Quantities Used to Describe Motion

1. To describe motion accurately, physicists use several related quantities:
   1. Distance
   2. Displacement
   3. Time
   4. Speed
   5. Velocity
   6. Acceleration
2. Each quantity provides different information about how an object moves.

Scalars and Vectors

1. A scalar quantity has magnitude only
2. A vector quantity has both magnitude and direction.

Distance vs Displacement

Distance

1. Distance describes how much ground an object covers during its motion, regardless of where it starts or finishes.
2. It increases as an object keeps moving, even if the object changes direction.
3. Taking a longer or winding path results in a greater distance than taking a short or direct path.
4. Distance does not tell us anything about direction, only about the total movement.

Displacement

1. Displacement focuses only on the starting and finishing points, not on the path between them.
2. It shows how far an object has moved and in which direction.
3. An object can travel a long distance but still have a small or even zero displacement.
4. Displacement includes direction and can be positive, negative, or zero.

Speed And Velocity Tell How Fast Position Changes

To discuss "how fast," you need to decide whether direction matters.

Speed

1. Speed describes how fast an object is moving.
2. Speed does not include information about direction.
3. Speed is therefore a scalar quantity.
4. The SI unit of speed is metres per second (m s⁻¹).
5. Speed is calculated by dividing the distance travelled by the time taken.

$$speed = \frac{distance}{time}$$

Velocity

1. Velocity tells us how fast an object is moving and in which direction.
2. Velocity is a vector quantity.
3. The unit of velocity is metres per second (m s⁻¹).
4. Velocity can be positive or negative depending on direction.
5. Velocity is calculated by dividing displacement by the time taken.

$$velocity = \frac{displacement}{time}$$

Average Speed and Average Velocity

$$\text{average speed} = \frac{\text{distance traveled}}{\Delta t}$$

$$\text{average velocity} = \frac{\Delta s}{\Delta t}$$

where $\Delta s$ is the change in displacement.

Acceleration

1. An object accelerates when:
   1. Its speed increases
   2. Its speed decreases (deceleration)
   3. Its direction changes
2. Acceleration is calculated by dividing the change in velocity by the time taken.
3. The unit of acceleration is metres per second squared (m s⁻²).

$$a = \frac{\Delta v}{t}$$

What Does A Displacement-Time Graph Show?

1. A displacement-time graph plots displacement (vertical axis) against time (horizontal axis). The key idea is:
2. Velocity is the rate of change of displacement, so on a displacement-time graph, the gradient (slope) gives the velocity.
3. Mathematically, over any interval, $$v = \frac{\Delta s}{\Delta t}$$
4. And on a graph, this is the slope of the line joining the two points.

What Straight Lines Mean

1. A horizontal line shows the object is stationary.
2. A straight sloping line shows constant velocity.
3. A steeper straight line shows faster constant velocity.
4. A line sloping downward shows motion in the opposite direction.

What Curves Mean

1. If a displacement-time graph is curved, then its gradient is changing, so the velocity is changing.
2. That means the object is accelerating (speeding up in one direction) or decelerating (slowing down).
   1. Curve getting steeper with time: speed is increasing.
   2. Curve getting less steep with time: speed is decreasing.

Finding Instantaneous Velocity With A Tangent

1. Instantaneous velocity can be determined by drawing a tangent to the curve at a specific point.
2. The gradient of the tangent gives the velocity at that instant.

Velocity-Time Graphs Show Acceleration Through The Gradient

1. A velocity–time graph plots velocity (vertical axis) against time (horizontal axis). A core relationship is:
2. The gradient of a velocity–time graph represents the acceleration.
3. Over an interval, $$a = \frac{\Delta v}{\Delta t}$$

Reading Velocity From The Graph

1. Positive velocity: motion in the positive direction.
2. Negative velocity: motion in the negative direction (sometimes described as "traveling backwards").
3. Zero velocity: momentarily at rest.

Reading Acceleration From The Graph

1. Horizontal line: zero acceleration, velocity is constant.
2. Straight sloping line: constant acceleration (positive slope) or constant deceleration (negative slope).

Area Under A Velocity–Time Graph Gives Displacement

1. The area under a velocity–time graph represents the distance travelled.
2. Larger areas correspond to greater distances.
3. Equal areas over equal time intervals indicate equal distances travelled.
   1. For constant velocity, the area is a rectangle.
   2. For constant acceleration (straight-line velocity–time graph), the area is a trapezium.

When Is Each Graph Most Useful? When Can They Be Used Together?

1. Displacement–time graphs are most useful for:
   1. Determining velocity from gradients
   2. Identifying whether motion is constant or changing
2. Velocity–time graphs are most useful for:
   1. Determining acceleration from gradients
   2. Finding distance from area
3. Together, these graphs provide a complete picture of motion.