Q. 51 The quantity dE/dv, the rate of ... [FREE SOLUTION]

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Chapter 36: Q. 51 (page 1061)

The quantity dE/dv, the rate of increase of energy with speed, is the amount of additional energy a moving object needs per $1 m / s$ increase in speed.
a. $A 25, 000 k g$ kg truck is traveling at $30 m / s$ . How much additional energy is needed to increase its speed by $1 m / s$ ?
b. $A 25, 000 k g$ rocket is traveling atrole="math" localid="1649677033976" $0.90 c$ . How much additional energy is needed to increase its speed by $1 m / s$ ?

Short Answer

Expert verified

a. $750 K J$ energy is needed to increase its speed by $1 m / s$ .

b. $8.2 脳 10^{13} J$ energy is needed to increase its speed by $1 m / s$

Step by step solution

Part (a) step 1: Given Information

We need to find the additional energy needed to increase its speed by $1 m / s$ in which his truck is traveling at $30 m / s$ .

Part (a) step 2: Simplify

Consider

$m = 25, 000 k g$

Here the speed is not relativistic $(v = 30 m / s)$

role="math" localid="1649678026406" $E = \frac{1}{2} m v^{2} \frac{d E}{d v} = m v \frac{d E}{d v} = (25, 000 k g) (30 m / s) \frac{d E}{d v} = 750 K J$ (Substitute values in the equation )

Part (b) step 1: Given Information

We need to find the additional energy needed to increase its speed by $1 m / s$ in witch rocket is traveling at $0.90 c$ .

Part (b) step 2: Simplify

Here the speed is definitely( $v = 0.90 c m / s$ ):

$\frac{d E}{d v} = \frac{d}{d v} (m c^{2} + K) \frac{d E}{d v} = \frac{d}{d v} (m c^{2} + 纬_{p} m c^{2} - m c^{2}) \frac{d E}{d v} = \frac{d}{d v} (纬_{p} m c^{2}) \frac{d E}{d v} = m c^{2} \frac{d}{d v} (1 - \frac{v^{2}}{c^{2}}) \frac{d E}{d v} = m c^{2} \frac{d}{d v} {(1 - \frac{v^{2}}{c^{2}})}^{- \frac{1}{2}} \frac{d E}{d v} = m c^{2} (- \frac{1}{2}) {(1 - \frac{v^{2}}{c^{2}})}^{- \frac{3}{2}} (\frac{- 2 v}{c^{2}}) \frac{d E}{d v} = m v {(1 - \frac{v^{2}}{c^{2}})}^{^{- \frac{3}{2}}} \frac{d E}{d v} = (25000 k g) (0.90) (3.0 脳 10^{8} m / s) {(1 - 0 . 9^{2})}^{- \frac{3}{2}} \frac{d E}{d v} = 8.2 脳 10^{13} J$

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Short Answer

Step by step solution

Part (a) step 1: Given Information

Part (a) step 2: Simplify

Part (b) step 1: Given Information

Part (b) step 2: Simplify

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