An object has a mass of $8 k g$ $8 kg$ . The object's kinetic energy uniformly changes from $240 K J$ $240 KJ$ to $640 K J$ $640KJ$ over $t \in [0, 3 s]$ $t in [0, 3 s]$ . What is the average speed of the object?

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An object has a mass of $8 k g$ $8 kg$ . The object's kinetic energy uniformly changes from $240 K J$ $240 KJ$ to $640 K J$ $640KJ$ over $t \in [0, 3 s]$ $t in [0, 3 s]$ . What is the average speed of the object?

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Answer 1 · 2017-06-02T18:18:38

The average speed is $= 328.7 m s^{- 1}$ $=328.7ms^-1$

Explanation:

The kinetic energy is

$K E = \frac{1}{2} m v^{2}$ $KE=1/2mv^2$

mass is $= 8 k g$ $=8kg$

The initial velocity is $= u_{1}$ $=u_1$

$\frac{1}{2} m u_{1}^{2} = 240000 J$ $1/2m u_1^2=240000J$

The final velocity is $= u_{2}$ $=u_2$

$\frac{1}{2} m u_{2}^{2} = 640000 J$ $1/2m u_2^2=640000J$

Therefore,

$u_{1}^{2} = \frac{2}{8} \cdot 240000 = 60000 m^{2} s^{- 2}$ $u_1^2=2/8*240000=60000m^2s^-2$

and,

$u_{2}^{2} = \frac{2}{8} \cdot 640000 = 160000 m^{2} s^{- 2}$ $u_2^2=2/8*640000=160000m^2s^-2$

The graph of $v^{2} = f (t)$ $v^2=f(t)$ is a straight line

The points are $(0, 60000)$ $(0,60000)$ and $(3, 160000)$ $(3,160000)$

The equation of the line is

$v^{2} - 60000 = \frac{160000 - 60000}{3} t$ $v^2-60000=(160000-60000)/3t$

$v^{2} = 33333.3 t + 60000$ $v^2=33333.3t+60000$

So,

$v = \sqrt{(33333.3 t + 60000)}$ $v=sqrt((33333.3t+60000)$

We need to calculate the average value of $v$ $v$ over $t \in [0, 3]$ $t in [0,3]$

$(3 - 0) ¯ v = \int_{0}^{3} \sqrt{(33333.3 t + 60000)} d t$ $(3-0)bar v=int_0^3sqrt((33333.3t+60000))dt$

$3 ¯ v = ⎡ ⎢ ⎣ {⎛ ⎝ \frac{{(33333.3 t + 60000)}^{\frac{3}{2}}}{\frac{3}{2} \cdot 33333.3} ⎤ ⎦}_{0}^{3}$ $3 barv=[((33333.3t+60000)^(3/2)/(3/2*33333.3)]_0^3$

$= ⎛ ⎝ \frac{{(33333.3 \cdot 3 + 60000)}^{\frac{3}{2}}}{50000} ⎞ ⎠ - ⎛ ⎝ \frac{{(33333.3 \cdot 0 + 60000)}^{\frac{3}{2}}}{50000} ⎞ ⎠$ $=((33333.3*3+60000)^(3/2)/(50000))-((33333.3*0+60000)^(3/2)/(50000))$

$= \frac{160000^{\frac{3}{2}}}{50000} - \frac{60000^{\frac{3}{2}}}{50000}$ $=160000^(3/2)/50000-60000^(3/2)/50000$

$= 986.1$ $=986.1$

So,

$¯ v = \frac{986.1}{3} = 328.7 m s^{- 1}$ $barv=986.1/3=328.7ms^-1$

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An object has a mass of $8 k g$ $8 kg$ . The object's kinetic energy uniformly changes from $240 K J$ $240 KJ$ to $640 K J$ $640KJ$ over $t \in [0, 3 s]$ $t in [0, 3 s]$ . What is the average speed of the object?

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Humanities

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An object has a mass of 8kg8 kg. The object's kinetic energy uniformly changes from 240KJ240 KJ to 640KJ 640KJ over t∈[0,3s]t in [0, 3 s]. What is the average speed of the object?

1 Answer

Explanation:

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An object has a mass of $8 k g$ $8 kg$ . The object's kinetic energy uniformly changes from $240 K J$ $240 KJ$ to $640 K J$ $640KJ$ over $t \in [0, 3 s]$ $t in [0, 3 s]$ . What is the average speed of the object?