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Motion in A Plane

The relation between time t and distance x is t=ax² +bx where a and b are constants. The acceleration is

−2abv²
2bv³
−2av³
2av²

−2av³

if t = ax²+bx

then, differentiate both side with respect to time (t)

\frac{d t}{d t} = 2 a x \frac{d x}{d t} + b \frac{d x}{d t} 1 = 2 a x v + b v v . (2 a x + b) = 1 (2 a x + b) = 1 / v

Again differentiate both side with respect to time (t)

2 a \frac{d x}{d t} = - v^{2} . \frac{d v}{d t} 2 a v = - v^{- 2} . a c c e l e r a t i o n a c c e l e r a t i o n = - 2 a v^{3}

Hence, retardation= -2av³

A car travels along a straight line for first half time with speed 40 km/hr and the second half time with speed 60 km/hr. Find the mean speed of the car?

Mean speed of car,

$straight v subscript straight m space equals space fraction numerator straight v subscript 1 straight t subscript 1 space plus space straight v subscript 2 straight t subscript 2 over denominator straight t subscript 1 space plus space straight t subscript 2 end fraction space space space space space space space equals space fraction numerator 40 space straight x space left parenthesis straight t divided by 2 right parenthesis space plus space 60 space left parenthesis straight t divided by 2 right parenthesis over denominator left parenthesis straight t divided by 2 space plus space straight t divided by 2 right parenthesis end fraction space space space space space equals space 50 space km divided by hr$

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The radius of the earth's orbit around the sun is 1.5 x 10¹¹ m. Calculate the angular and linear velocity of the earth?

Given,

Radius of Earth's orbit, r = 1.5 x 10¹¹ m

Time period of revolution of earth around the sun is 1 year.

That is,

T = 1 year = 365 x 24 x 60 x 60 s

Therefore,

Angular velocity, $straight omega space equals space fraction numerator 2 straight pi over denominator straight T end fraction$

$equals space fraction numerator 2 space cross times space open parentheses begin display style bevelled 22 over 7 end style close parentheses over denominator 365 space straight x space 24 space straight x space 60 space straight x 60 end fraction space equals space 1.99 space cross times space 10 to the power of negative 7 end exponent space sec$

Linear velocity, v = $straight omega space straight r$

$equals space 1.99 space cross times space 10 to the power of negative 7 end exponent space cross times space 1.5 space cross times space 10 to the power of 11 equals space 2.99 space cross times space 10 to the power of 4 space straight m divided by straight s$

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Two persons P and Q are standing 54 m apart on a long moving belt. Person P rolls a round stone towards a person Q with a speed of 9 m/s with respect to belt. If the belt is moving with a speed of 4 m/s in the direction from P to Q. What is the speed of the stone w.r.to an observer on a stationary platform?

Let the distance from P to Q be positive.

Given, speed of the belt, v_b = +4 m/s

Speed of the stone w.r.t belt, v_s = + 9 m/s

Speed of the stone w.r.t a stationary observer= v_s + v_b = 9+ 4 = 13 m/s

215 Views

A plane flying horizontally at 100 m/s at a height of 1000 m releases a bomb from it. Find the time it takes to reach the ground.

We have,

Initial distance, y_o = 0

Height attained after time t, y = 1000 m

Initial velocity, u_y = 0 m/s

Acceleration, a_y = 9.8 m/s²

Time = t

Now, using the equation of motion, we have

$space space space space straight y space equals space straight y subscript straight o space plus space straight u subscript straight y straight t space plus space 1 half straight a subscript straight y straight t squared space therefore space 1000 space equals space 0 space plus space 0 space left parenthesis straight t right parenthesis space plus space 1 half cross times 9.8 cross times space straight t squared space space space space space space space space space space space space space equals space 4.9 space straight t squared space rightwards double arrow space straight t space equals space square root of fraction numerator 1000 over denominator 4.9 end fraction end root space equals space 100 over 7 space equals space 14.28 space sec$

Therefore, 14.28 sec ia taken for the bomb to reach the ground.

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Motion in A Plane

The radius of the earth's orbit around the sun is 1.5 x 10¹¹ m. Calculate the angular and linear velocity of the earth?

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Motion in A Plane

The radius of the earth's orbit around the sun is 1.5 x 1011 m. Calculate the angular and linear velocity of the earth?

The radius of the earth's orbit around the sun is 1.5 x 10¹¹ m. Calculate the angular and linear velocity of the earth?