3.4 Motion with Constant Acceleration

Kinematics with constant acceleration is all about understanding how objects move when their speed changes at a steady rate. It's like tracking a car speeding up or slowing down on a straight road.

We'll look at the math behind this motion, using equations that connect speed, distance, time, and acceleration. These tools help us predict where objects will be and how fast they'll go, which is super useful in real-world situations like traffic flow or sports.

Kinematics with Constant Acceleration

Vector and Scalar Quantities in Motion

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• Understand the difference between vector and scalar quantities in motion
  • Vector quantities have both magnitude and direction (e.g., velocity, acceleration, displacement)
  • Scalar quantities have only magnitude (e.g., speed, distance, time)
• Recognize the importance of reference frames when describing motion
• Distinguish between average acceleration and instantaneous acceleration

Kinematic equations for constant acceleration

• Understand and apply the four kinematic equations for constant acceleration:
  • $v = v_0 + at$ calculates final velocity $v$ using initial velocity $v_0$, acceleration $a$, and time $t$
  • $x = x_0 + v_0t + \frac{1}{2}at^2$ calculates displacement $x$ using initial position $x_0$, initial velocity $v_0$, acceleration $a$, and time $t$
  • $v^2 = v_0^2 + 2a(x - x_0)$ relates final velocity $v$, initial velocity $v_0$, acceleration $a$, and displacement $(x - x_0)$
  • $x = \frac{1}{2}(v_0 + v)t$ calculates displacement $x$ using average velocity $\frac{1}{2}(v_0 + v)$ and time $t$
• Identify given variables (initial position, initial velocity, acceleration, time) in a problem and select the appropriate equation
• Solve for the unknown variable using algebraic manipulation of the chosen equation
• Consider the sign of acceleration: positive for acceleration in the direction of motion, negative for deceleration or acceleration opposite to motion
• Maintain consistent units (m, s, m/s, m/s²) throughout problem-solving, converting if necessary

Calculations for one-dimensional motion

• Calculate displacement $(\Delta x)$, the change in an object's position
  • $\Delta x = x - x_0$ is final position minus initial position, can be positive or negative based on direction (left or right)
• Determine velocity $(v)$, the rate of change of position with respect to time
  • Average velocity $v_{avg} = \frac{\Delta x}{\Delta t}$ is total displacement divided by total time
  • Instantaneous velocity $v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}$ is velocity at a specific instant, found by taking the limit as time interval approaches zero
• Find time $(t)$, the duration of motion, using kinematic equations when given displacement, initial velocity, and acceleration (car traveling for 5 s, ball falling for 2.3 s)

Analysis of pursuit scenarios

• Solve pursuit problems involving two objects moving in the same direction, with one trying to catch the other (police car chasing a speeding vehicle, dog running to catch a ball)
• Identify positions, velocities, and accelerations of both pursuer and pursued objects
• Apply appropriate kinematic equations to set up equations for each object's motion
• Equate positions of the objects at the moment of catch-up to solve for the unknown variable, typically time
• Analyze relative velocity between objects: pursuer's velocity minus pursued object's velocity
  • The distance between the objects decreases at the rate of the relative velocity
• Determine if catch-up is possible based on given conditions (maximum velocities, accelerations, initial distances)
  • If pursuer's maximum velocity is less than the pursued object's velocity, catch-up is impossible