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Natalka [10]
3 years ago
15

A particle moves on a line away from its initial position so that after t hours it is s = 6t2 + 2t miles from its initial positi

on. find the average velocity of the particle over the interval [1, 4]. include units in your answer.
Physics
1 answer:
docker41 [41]3 years ago
6 0
<span>32 mph
   First, let's calculate the location of the particle at t=1, and t=4
 t=1
 s = 6*t^2 + 2*t
 s = 6*1^2 + 2*1
 s = 6 + 2
 s = 8 t = 4
 s = 6*t^2 + 2*t s = 6*4^2 + 2*4
 s = 6*16 + 8
 s = 96 + 8
 s = 104
   So the particle moved from 8 to 104 over the time period of 1 to 4 hours. And the average velocity is simply the distance moved over the time spent. So:
 avg_vel = (104-8)/(4-1) = 96/3 = 32
   And since the units were miles and hours, that means that the average speed of the particle over the interval [1,4] was 32 miles/hour, or 32 mph.</span>
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Answer:

The coefficient of friction between the cars and the road is 0.859.

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The two cars collide each other inelastically, then we can determine the resulting velocity by the Principle of Momentum Conservation:

m_{A}\cdot v_{A} + m_{B}\cdot v_{B} = (m_{A} + m_{B})\cdot v (1)

Where:

m_{A}, m_{B} - Masses of the cars, in kilograms.

v_{A}, v_{B} - Initial velocities of the cars, in meters per second.

v - Velocity of the resulting system, in meters per second.

If we know that m_{A} = 2120\,kg, v_{A} = 13.4\,\frac{m}{s }, m_{B} = 2810\,kg and v_{B} = 0\,\frac{m}{s}, then the  velocity of the resulting system:

v = \frac{m_{A}\cdot v_{A}+m_{B}\cdot v_{B}}{m_{A}+m_{B}}

v = \frac{(2120\,kg)\cdot \left(13.4\,\frac{m}{s} \right)+(2810\,kg)\cdot \left(0\,\frac{m}{s} \right)}{2120\,kg + 2810\,kg}

v = 5.762\,\frac{m}{s}

By Principle of Energy Conservation and Work-Energy Theorem, we understand that the initial translational kinetic energy (K), in joules, is dissipated due to work done by friction (W_{f}), in joules, that is to say:

K = W_{f} (2)

\frac{1}{2}\cdot (m_{A}+m_{B})\cdot v^{2} = \mu\cdot (m_{A}+m_{B})\cdot g \cdot s

\frac{1}{2}\cdot v^{2} = \mu \cdot g\cdot s (2b)

Where:

\mu - Coefficient of friction, no unit.

g - Gravitational acceleration, in meters per square second.

s- Travelled distance, in meters.

If we know that v = 5.762\,\frac{m}{s}, g = 9.807\,\frac{m}{s^{2}} and s = 1.97\,m, then the coefficient of friction is:

\mu = \frac{v^{2}}{2\cdot g\cdot s}

\mu = \frac{\left(5.762\,\frac{m}{s} \right)^{2}}{2\cdot \left(9.807\,\frac{m}{s^{2}} \right)\cdot (1.97\,m)}

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