#Introduction to Straight Line Motion

#Table of Contents

1. What is Straight Line Motion?
2. Terminology
   • Particle
   • Time
   • Displacement
   • Distance
   • Velocity
   • Speed
   • Acceleration
3. Velocity & Acceleration as Derivatives
4. Worked Example
5. Practice Questions
6. Glossary
7. Summary and Key Takeaways

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#What is Straight Line Motion?

Straight line motion models how objects move in a straight line with respect to time.

• This can be described as motion along the $$x$$-axis.
• The straight line will have both a positive and a negative direction.
  • On the $$x$$-axis, this will be the usual positive and negative directions.
  • If not specified, you can choose the positive and negative directions but remain consistent with your choice.

#Terminology

#Particle

• A particle is a term used for an object.
• It is assumed to be the size of a single point, so its 3D dimensions are not considered.

#Time ($t$)

• Time is usually measured in seconds ($$\mathrm{s}$$).
• Displacement, velocity, and acceleration are all functions of time ($$t$$).
• The term 'initial' or 'initially' means when $$t = 0$$.

#Displacement ($s$)

• $$s$$ is the standard notation for displacement.
  • For motion along the $$x$$-axis, $$x$$ may be used instead of $$s$$.
• Displacement is typically measured in meters ($$\mathrm{m}$$) or feet ($$\mathrm{ft}$$).
  • Sometimes, questions may use "units" without specifying a particular unit.
• The displacement of a particle is its distance relative to a fixed point.
  • This fixed point may be (but is not always) the particle’s initial position.
• Displacement is zero when the particle is at the fixed point ($$s = 0$$).
  • Positive if the particle is in the positive direction from the fixed point.
  • Negative if it is in the negative direction from the fixed point.

#Distance ($d$)

• Distance is the magnitude of displacement.
• It can refer to:
  • The distance traveled by a particle.
  • The straight-line distance from a particular point.
• Distance is always positive (or zero).

#Velocity ($v$)

• Velocity is usually measured in feet or meters per second.
• It is the rate of change of displacement at time $$t$$: $$v = \frac{ds}{dt}$$
  • Positive if the particle is moving in the positive direction.
  • Negative if moving in the negative direction.
• If the particle is stationary, its velocity is zero ($$v = 0$$).

#Speed

• Speed is the magnitude (absolute value or modulus) of velocity.
  • For example, if $$v = 4$$, speed $$= |4| = 4$$
  • If $$v = -6$$, speed $$= |-6| = 6$$

#Acceleration ($a$)

• Acceleration is usually measured in feet or meters per second squared.
  • It is the rate of change of velocity at time $$t$$: $$a = \frac{dv}{dt}$$
  • Positive or negative, but the sign alone does not fully describe motion.
  • If velocity and acceleration have the same sign, the particle is accelerating (speeding up).
  • If they have different signs, the particle is decelerating (slowing down).
• When acceleration is zero ($$a = 0$$), the particle moves with constant velocity.

**Note**: The direction of motion is determined by the sign (\+ or \-) of the velocity.

#Velocity & Acceleration as Derivatives

#What is Velocity as a Derivative?

• Velocity is the rate of change of displacement: $$v = \frac{ds}{dt}$$
  • Differentiate displacement to get velocity.
• Velocity is the slope of a displacement-time graph.

#What is Acceleration as a Derivative?

• Acceleration is the rate of change of velocity: $$a = \frac{dv}{dt}$$
  • Differentiate velocity to get acceleration.
• Acceleration is the slope of a velocity-time graph.
• It is also the rate of change of the rate of change of displacement: $$a = \frac{d^2 s}{dt^2}$$
  • Differentiate displacement twice to get acceleration.

#Worked Example

The displacement from the origin of a particle $$P$$, as it travels along the $$x$$-axis, is given by: $$s_P = 3 \sin\left(\frac{1}{2} t \right)$$

The displacement from the origin of a second particle $$Q$$ is given by: $$s_Q = \frac{1}{8} t^3 - \frac{7}{4} t^2 + 5t$$

$$s_P$$ and $$s_Q$$ are measured in meters, and $$t$$ is measured in seconds for $$0 \le t \le 10$$.

(a) Determine which particle is furthest from the origin at $$t = 5$$

Practice Question

Question: Find the displacement of each particle at $$t = 5$$.

Answer:

For particle $$P$$: $$s_P = 3 \sin\left(\frac{1}{2} \cdot 5\right) \approx 1.7954 \text{ meters}$$

For particle $$Q$$: $$s_Q = \frac{1}{8} (5)^3 - \frac{7}{4} (5)^2 + 5 \cdot 5 = -3.125 \text{ meters}$$

Therefore, particle $$Q$$ is furthest from the origin at $$t = 5$$.

(b) At $$t = 5$$, determine if the particles are moving closer together, or further apart. Explain your reasoning in your working.

Practice Question

Question: Find the velocities of each particle at $$t = 5$$.

Answer:

For particle $$P$$: $$v_P = \frac{d s_P}{dt} = \frac{3}{2} \cos\left(\frac{1}{2} t\right)$$ $$v_P = \frac{3}{2} \cos\left(\frac{1}{2} \cdot 5\right) \approx -1.2017 \text{ meters per second}$$

For particle $$Q$$: $$v_Q = \frac{d s_Q}{dt} = \frac{3}{8} t^2 - \frac{7}{2} t + 5$$ $$v_Q = \frac{3}{8} (5)^2 - \frac{7}{2} \cdot 5 + 5 = -3.125 \text{ meters per second}$$

Since both particles have negative velocities, they are moving to the left. Therefore, the particles are moving further apart.

(c) At $$t = 5$$, determine which particle has the greatest magnitude of acceleration.

Practice Question

Question: Find the accelerations of each particle at $$t = 5$$.

Answer:

For particle $$P$$: $$a_P = \frac{d v_P}{dt} = -\frac{3}{4} \sin\left(\frac{1}{2} t\right)$$ $$a_P = -\frac{3}{4} \sin\left(\frac{1}{2} \cdot 5\right) \approx -0.4489 \text{ meters per second squared}$$

For particle $$Q$$: $$a_Q = \frac{d v_Q}{dt} = \frac{3}{4} t - \frac{7}{2}$$ $$a_Q = \frac{3}{4} \cdot 5 - \frac{7}{2} = 0.25 \text{ meters per second squared}$$

While the acceleration of $$P$$ is negative, it has the greatest magnitude.

#Practice Questions

#Multiple Choice Questions

1. What is the standard notation for displacement?
   • (a) $$t$$
   • (b) $$s$$
   • (c) $$v$$
   • (d) $$a$$
2. If a particle's velocity is zero, what does this imply about its motion?
   • (a) It is accelerating.
   • (b) It is decelerating.
   • (c) It is stationary.
   • (d) It is moving in a negative direction.

#Short Answer Questions

1. Explain the difference between displacement and distance.
2. How do you determine the velocity of a particle from its displacement-time graph?

#Glossary

• Particle: An object considered to be a single point in space.
• Displacement: The distance and direction of an object's position relative to a fixed point.
• Velocity: The rate of change of displacement.
• Acceleration: The rate of change of velocity.
• Speed: The magnitude of velocity, ignoring direction.
• Distance: The magnitude of displacement, always positive.

#Summary and Key Takeaways

#Summary

• Straight line motion describes how objects move in a straight line over time.
• Key variables include displacement ($$s$$), velocity ($$v$$), and acceleration ($$a$$).
• Velocity and acceleration can be determined as derivatives of displacement and velocity, respectively.

#Key Takeaways

• Displacement is the distance relative to a fixed point and can be positive or negative.
• Distance is always positive and is the magnitude of displacement.
• Velocity indicates the rate of change of displacement and can be positive or negative.
• Speed is the magnitude of velocity.
• Acceleration indicates the rate of change of velocity and can be positive or negative.