Kinematics: Motion Analysis

Mechanics - Why?

• A question from the 2005 PHSI 110 final exam involved two cars, A and B.
• Car A travels at a constant velocity of 15 m/s and passes stationary Car B at time t = 0.
• At the time of passing, Car B starts moving with a constant acceleration of 0.1 m/s² in the same direction as Car A.
• The question asks to find the speed of Car B when it passes Car A.
• In 2005, less than 50% of the students in PHSI110 answered this question correctly.

Kinematics - Learning Goals

• Kinematics is the study of motion.
• Fundamental kinematic quantities include time, distance, speed, and acceleration.
• Vector quantities include displacement, velocity, and acceleration.
• Kinematics also deals with average and instantaneous quantities.
• Relative displacement, velocity, and acceleration are also key concepts.

Kinematics

• Kinematics involves the description of motion and identifying the necessary quantities to adequately describe it.
• Motion involves a change in position with respect to time.
• It is necessary to measure both time and position to understand motion.
• Rates of change, specifically position, are important.
• The rate of change of the rate of change is also a key concept.

Time, Distance, Speed

• Speed is the rate of change of an object's position over time.
• This involves the formula v = Δx/Δt, where Δ represents "change in."
• To determine the distance Δx an object travels at a constant speed v over a time interval Δt, use the formula Δx = vΔt.
• For example, a car moving at 5 m/s for 3 seconds travels a distance of Δx = 5 m/s * 3 s = 15 m.

Average Speed

• Average speed is the total distance traveled divided by the total time taken.
• The equation for average speed is vav = Δx/Δt, where Δx is the total distance and Δt is the total time.
• The formula calculates the average speed but it does not represent the speed of an object at a specific moment.
• For constant acceleration, average speed can also be calculated as vav = (vf + vi)/2.

Instantaneous Speed

• Instantaneous speed refers to the speed of an object at a particular moment in time.
• The average speed approaches the instantaneous speed as the changes in position Δx and time Δt become smaller.

Example

• If someone drives 300 km to Christchurch in 5 hours, the average speed is 60 km/h.
• Average speed does not account for variations during the trip.
• Instantaneous speed is the speed at a precise moment (e.g., at exactly 12:30).

Acceleration

• Acceleration occurs when an object's speed changes by a certain amount every second.
• For example, a car starting from rest and increasing its speed by 2 m/s every second has an acceleration of 2 m/s².
• Uniform (or linear) acceleration means the change in velocity per second remains constant.

Acceleration

• Acceleration is the rate of change of speed.
• The formula for acceleration is a = Δv/Δt.
• The change in speed with a given uniform acceleration over time is Δv = aΔt.
• For a car accelerating from rest for 4 seconds at 2 m/s², the change in speed is ∆v = 2 m/s² * 4 s = 8 m/s.

Example

• A train increases speed uniformly from 3 m/s to 11 m/s over 100 seconds.
• Average speed of the train is Vav = (11 + 3) / 2 = 7 m/s.
• The train travels ∆x = 7 m/s * 100 s = 700 m.

Uniform Acceleration

• Equations apply to uniform acceleration.
• Distance (Ax) is expressed as Ax = vav∆t = 1/2(vi + vf)∆t = vit + 1/2a².
• Distance when starting from rest: Ax = 1/2a².

Vector Quantities

• Displacement is a vector describing the distance and direction to an object, (e.g., 3 m north, 4 m east).
• Velocity is the rate of change of displacement and includes direction, (e.g., 12 km/h south, 10 m/s NE).
• Acceleration is the rate of change of velocity.
• Deceleration is acceleration opposite to the velocity vector.

Adding Velocities

• Relative velocities of an airplane flying in and against the wind are considered.
• The resultant velocity is the sum or difference of the airplane's speed and the wind speed depending on the direction.

Adding Velocities

• Calculating the resultant velocity of a bird flying north at 3 m/s with a 4 m/s wind blowing east.
• The resultant velocity calculation consists of d2 = 32 + 42 = 25, results to d = 5 m
• tan θ = opposite / adjacent = 3/4, leads to θ = 37° N. of E.
• The resultant velocity is calculated as 5 m/s at 37° N of E.

Acceleration due to Gravity

• Galileo discovered that all objects falling freely toward Earth have the same acceleration
• Every object in free fall increases its downward speed by 9.8 m/s every second (a = g = 9.8 m/s² ≈ 10 m/s²)
• g is known as the acceleration due to gravity.
• The equation ∆v = at simplifies to ∆v = g∆t under gravity.
• For example, after 5 seconds of free fall from rest, ∆v = g∆t = 10 m/s² * 5 s = 50 m/s

Example

• A cricket ball is dropped from a plane.
• Velocity after 5 seconds is 50 m/s.
• The start velocity is 0 m/s.
• The average velocity is Vav = (0 + 50) / 2 = 25 m/s.
• This means distance fallen in 5 s is 25 m/s * 5 s = 125 m.
• Distance fallen in 5 s is also calculated Ax = 1/2 gAt² which comes to Ax = ½ * 10 * 52 = 125 m.

Example

• Throwing a cricket ball straight up at 30 m/s.
• The gravity will reduce its upward velocity by 10 m/s every second.
• When its upward velocity is zero, it will reach its highest point.
• The time to reach the highest point is At = 30 m/s / 10 m/s² = 3 s.
• During this time, the average velocity is (30 + 0) / 2 = 15 m/s.
• Distance is Ax = 15 m/s * 3 s = 45 m.

Example

• The time it takes to reach the ground again after being thrown upwards will be the same time it takes to come down. .
• vi = 0, Ax = 45 m, g = 10 m/s² : Ax = 1/2 gAt2.
• 45 = 1/2 x 10 x At2 = 5At2.
• At2 = 9* or At = 3 s
• The downward velocity at the ground is v = 10 m/s2 * 3 s = 30 m/s.

Vertical and Horizontal Motion

• Vertical and horizontal motions are independent.
• An image of two golf balls shows one released to fall and the other projected horizontally.

Example - Relative Motion

• If a polar bear is 200m behind a man running at 7 m/s and the bear runs at 11 m/s
• The time it will take the bear to catch the man is 50 seconds.