3.10

Kinematic Equations: Problem Solving

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The kinematic equations of motion are useful to solve problems involving one-dimensional motion of objects under constant acceleration.

Consider a couple driving to a nearby coffee shop. They start the car and apply a constant acceleration of 2 meters per second squared. What will be the car's velocity after 20 seconds and the distance covered by it in that time?

The choice of the equation to solve the problem depends on the known quantities and the unknown quantities.

Here, the known quantities are constant acceleration, time, the initial position, and the initial velocity, as the car was at rest. The unknown quantities are the velocity and distance covered after 20 seconds, which can be calculated using the first and second kinematic equations.

Substituting known values in the first kinematic equation gives the velocity of the car, which equals 40 meters per second.

Then, substituting the known values in the second kinematic equation, simplifying and solving it gives the distance covered by the car equal to 400 meters.

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When analyzing one-dimensional motion with constant acceleration, the problem-solving strategy involves identifying the known quantities and choosing the appropriate kinematic equations to solve for the unknowns. Either one or two kinematic equations are needed to solve for the unknowns, depending on the known and unknown quantities. Generally, the number of equations required is the same as the number of unknown quantities in the given example. Two-body pursuit problems always require two equations to be solved simultaneously to derive the value of the unknown.

In complex problems, it is not always possible to identify the unknowns or the order in which they should be calculated. In such scenarios, it is useful to make a list of unknowns and draw a sketch of the problem to identify the directions of motion of an object. To solve the problem, substitute the knowns along with their units into the appropriate equation. This step provides a numerical answer, and also provides a check on units that can help find errors. If the units are incorrect, then an error has been made. However, correct units do not necessarily guarantee that the numerical part of the answer is also correct.

The final step in solving problems is to check the answer to see if it is reasonable. This final step is crucial as the goal of physics is to describe nature accurately. To see if the answer is reasonable, check both its magnitude and its sign, in addition to its units. This enables us to get a conceptual understanding of the problems that are being solved. Sometimes the physical principle may be applied correctly to solve the numerical problem, but produces an unreasonable result. For example, if an athlete starting a foot race accelerates at 0.4 m/s² for 100 seconds, their final speed will be 40 m/s (about 150 km/h). This result is unreasonable because a person cannot run at such a high speed for 100 seconds. Here, the physics is correct in a sense, but there is more to describing nature than just manipulating equations correctly.

This text is adapted from Openstax, University Physics Volume 1, Section 3.4: Motion with Constant Acceleration.

Transcript

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Kinematic Equations

Problem Solving

Constant Acceleration

One Dimensional Motion

Two Body Pursuit

Answer Verification

Motion Sketching

Unknown Quantities

Equation Selection