An object has a mass of 4 kg. The object's kinetic energy uniformly changes from 24 KJ to 42KJ over t in [0, 9 s]. What is the average speed of the object?

2 Answers
Mar 31, 2017

4 m/s

Explanation:

Let the rate of increase of kinetic energy with time be some m (KJ)/s where m is a constant as given in the question.

therefore (dE(t))/(dt) = m where E is Kinetic Energy and t is time.

implies dE(t)=m*dt
implies intdE = int m*dt

implies E(t) = m*t + c where c is a constant of integration.
We have two conditions:-
1. E(0) = 24 &
2. E(9) = 42

from 1. & 2.
24 = c & 42 = 9*m + 24 implies m=2
therefore E(t) = 2*t + 24

implies (1/2)*m*v^2 = 2*t+24 where m is the mass & v is the velocity (speed) of body.

implies v^2 = t+12
implies v = sqrt(t+12)

Average speed = (int_(t_1)^(t_2)v*dt)/(int_(t_1)^(t_2)dt)
where
t_1 = 0 & t_2=9

therefore Avg. speed = (int_(0)^(9)sqrt(t+12)*dt)/(int_(0)^(9)dt)
= (2/3*(t+12)^(3/2))/(t)]_0^9 = 2/3*(21sqrt21 - 12sqrt12)/9 approx 4

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Mar 31, 2017

128.05 ms^-1, rounded to two decimal places.

Explanation:

Let rate of change of kinetic energy be
(dKE(t))/(dt) = C
where C is constant with time t.

Integrating both sides with t we get
int (dKE(t))/(dt) cdot dt = int C cdotdt

=>KE= Ccdott + c ......(1)
where c is a constant of integration.

To evaluate c, use initial condition at t=0
24xx10^3 = Ccdot0+c
=>c=24xx10^3
We have expression for kinetic energy as
KE= Ccdott + 24xx10^3

To evaluate C, use final condition at t=9
42 xx10^3= Ccdot9+24xx10^3
=>C=2xx10^3

Equation (1) becomes
KE= (2cdott + 24)xx10^3 ......(2)

If m is the mass and v(t) is the velocity of body, Kinetic energy is given as
KE=1/2mv^2
=> 1/2mv^2(t) = (2*t+24)xx10^3
Given is m=4kg. Above expression becomes
=> 1/2xx4v^2(t) = (2*t+24)xx10^3
=> v(t)^2 = (t+12)xx10^3
=> v= +-sqrt((t+12)xx10^3)
=> "speed "|v|= sqrt((t+12)xx10^3)

Average speed ="Total Distance traveled"/"Time taken"=(int_0^9|v(t)|*dt)/(9-0)

:. Average speed = (int_(0)^(9)sqrt((t+12)xx10^3)*dt)/9
= sqrt(10^3)/9[(t+12)^(3/2)/(3/2)]_0^9
= (2sqrt(10^3))/27(21^(3/2) - 12^(3/2))
= 128.05 ms^-1, rounded to two decimal places.