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Wize University Dynamics Textbook (Master) > Kinematics: Rectilinear Motion

Non-Uniformly Accelerated Motion

Uniformly Accelerated MotionPrevious Section
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When the motion of a particle or a body is not undergoing uniform acceleration, we must return to the fundamental relationship between our 3 dynamics variables:

v=x˙=dxdtv = \dot{x}=\frac{dx}{dt}

a=v˙=dvdt=x¨=d2xdt2=vdvdxa = \dot{v}=\frac{dv}{dt}=\ddot{x} = \frac{d^2x}{dt^2}=v\frac{dv}{dx}

In these problems, acceleration may be given as a function of time, velocity or (rarely) position. You may also need to use initial conditions to determine the full form of your other functions (velocity and position).
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A car is accelerating from rest according to the following relationship:

a(t)=5/ta(t) = 5/t

At t = 1 second, the car's velocity is 3 m/s. Determine the velocity of the car at t = 5 seconds.


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a(t)=5ta\left(t\right)=\frac{5}{t}

t=1s, v=3t=1s,\ v=3

@ t=5→v=?@\ t=5\to v=?

Δx=?\Delta x=?

a=dvdt→v=∫5tdt=5ln⁡t+c→c=3a=\frac{dv}{dt}\to v=\int_{ }^{ }\frac{5}{t}dt=5\ln t+c\to c=3

v=5ln⁡5+3=11.05 msv=5\ln5+3=11.05\ \frac{m}{s}

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A common metric for comparing the power of different cars is the time they take to accelerate from 0 to 60 km/h. You're asked to evaluated two different designs for the exterior of a car, which effects its acceleration time. For both designs, the acceleration is proportional to the output power of the engine, minus a term to account for the frictional drag force (air resistance). The two acceleration functions are given below:

a1(v)=2−0.0005v2a_1 (v)= 2-0.0005v^2

a2(v)=23/(v+1)a_2(v)=23/(v+1)

What is the minimum length of track needed to test the two designs, if they both require 15 m of braking distance?


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The velocity of a particle along a straight line is given by the following function:

v(t)=2t2−16t+24v(t) = 2t^2-16t+24

where the velocity is in m/s and time is in seconds.

Determine:

a) The acceleration of the particle after 3 seconds
b) The displacement of the particle between t = 1 and t = 3 seconds
c) The distance the particle has travelled between t = 5 and t = 8 seconds


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Quiz: Non-Uniform Acceleration Practice Question
What is the distance travelled within the first 5 seconds of a particle whose motion along a line is described by the following equation:

v(t)=3t3−27tv(t) = 3t^3-27t