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Kinematics
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Computations involving distance, displacement, velocity, and acceleration
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Displacement vs Distance
AHL AI 5.13
Displacement s is the change in position between start and end time, whereas distance d is the total length of the path taken.
Direction does not matter for distance, which is never negative, but displacement can be negative - usually indicating motion down or to the left.
Displacement, velocity and acceleration
AHL AI 5.13
Acceleration is the rate of change of velocity, which is the rate of change of displacement.
v=dtds🚫\\
a=dtdv=dt2d2s📖
Hence, the integral of acceleration is velocity, and the integral of velocity is displacement.
Average Velocity and Acceleration
AHL AI 5.13
While we use derivatives to get instantaneous velocity and time, we can also find average velocity and time:
average velocity=ΔtΔs,average acceleration=ΔtΔv.
Change in Displacement from Velocity
AHL AI 5.13
Change in displacement between t1,t2:
∫t1t2v(t)dt📖
Speed
AHL AI 5.13
Speed is the magnitude of velocity:
speed=∣v∣
Speeding Up and Slowing Down
AHL AI 5.13
If the velocity and acceleration of an object point in the same direction, we say the object is "speeding up" since its speed increases.
If the velocity and acceleration of an object point in opposite directions, we instead say the object is "slowing down."
Distance from Velocity
AHL AI 5.13
The distance can be found from the velocity using the equation
∫t1t2∣v(t)∣dt📖
Since speed is given by ∣v(t)∣, we see that distance is the integral of speed.
Velocity as a vector and 2D motion
AHL AI 3.12
We can expand kinematics to two dimensions by thinking of displacement, velocity, and acceleration as 2D vectors, usually with x and y components. Then, velocity is given by
v⃗=(x′(t)y′(t)).
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