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Acceleration refers to how quickly an object’s velocity changes over time. It is a vector quantity, meaning it has both magnitude and direction. In the context of kinematics, understanding acceleration is key to solving problems where an object’s speed changes.
For our problem, the average acceleration is given as 0.640 m/s² directed due west. This tells us how much the runner's velocity increases per second in a specific direction.
When calculating acceleration, we often use the formula: \[ a = \frac{\Delta v}{t} \]where \(a\) is the average acceleration, \(\Delta v\) is the change in velocity, and \(t\) is the time over which the change occurs. Since acceleration here is positive, this implies that the object is speeding up in the direction specified, which is due west in this case.

Velocity is a measure of both the speed and direction of an object’s motion. Unlike speed, velocity is a vector, meaning it accounts for direction. This distinction is crucial for solving kinematics problems in physics.
In the original exercise, we are told that the final velocity of the runner is 4.15 m/s due west. This velocity indicates not only how fast the runner is moving but also the direction of the movement.
To find the initial velocity when the runner began accelerating, we can rearrange the equation for acceleration:\[ v_i = v_f - (a \times t) \] where \(v_i\) is the initial velocity, \(v_f\) is the final velocity, \(a\) is acceleration, and \(t\) is time. Using this formula, students can plug in the numbers to find how fast the runner was moving at the start of the acceleration.

Time is an integral component in kinematics, representing the duration over which motion occurs or changes. In problems involving kinematics, time helps determine how other variables like velocity and acceleration interact.
In the given problem, the time span during which the runner accelerates is 1.50 seconds. Knowing the duration of this phase allows us to calculate the change in velocity with given acceleration.
Time, denoted by \(t\), is the period during which all these changes are measured. In equations, remember to keep the units consistent; here, time is measured in seconds. Proper understanding of how time fits into kinematic equations enables students to link various aspects of motion effectively. During calculations, accurately accounting for time is crucial, as even slight errors in the time measure can significantly affect the outcome of kinematic equations.