A particle moves along the x-axis and has its displacement x varying with time t according to the equation where and c are constants of appropriate dimensions. Then, which of the following statements is correct? [2025]

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Q.
A particle moves along the x-axis and has its displacement x varying with time t according to the equation $x = c_{0} (t^{2} - 2) + c {(t - 2)}^{2}$ where $c_{0}$ and c are constants of appropriate dimensions. Then, which of the following statements is correct [2025]

Solution

Q.
A particle moves along the x-axis and has its displacement x varying with time t according to the equation $x = c_{0} (t^{2} - 2) + c {(t - 2)}^{2}$ where $c_{0}$ and c are constants of appropriate dimensions. Then, which of the following statements is correct [2025]

1 the acceleration of the particle is $2 c_{0}$

2 the acceleration of the particle is 2c

3 the initial velocity of the particle is 4c

4 the acceleration of the particle is $2 (c + c_{0})$

Ans.
(4)

$x = C_{0} (t^{2} - 2) + C (t^{2} + 4 - 4 t)$

$v = \frac{d x}{d t} = 2 t C_{0} + 2 C (t - 2)$

$a = \frac{d v}{d t} = 2 C_{0} + 2 C$

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Solution

Q. A particle moves along the x-axis and has its displacement x varying with time t according to the equation x=c0(t2–2)+c(t–2)2 where c0 and c are constants of appropriate dimensions. Then, which of the following statements is correct [2025]

1 the acceleration of the particle is 2c0

2 the acceleration of the particle is 2c

3 the initial velocity of the particle is 4c

4 the acceleration of the particle is 2(c+c0)

Ans. (4) x=C0(t2–2)+C(t2+4–4t) v=dxdt=2tC0+2C(t–2) a=dvdt=2C0+2C

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Q.
A particle moves along the x-axis and has its displacement x varying with time t according to the equation $x = c_{0} (t^{2} - 2) + c {(t - 2)}^{2}$ where $c_{0}$ and c are constants of appropriate dimensions. Then, which of the following statements is correct [2025]

1 the acceleration of the particle is $2 c_{0}$

4 the acceleration of the particle is $2 (c + c_{0})$

Ans.
(4)

$x = C_{0} (t^{2} - 2) + C (t^{2} + 4 - 4 t)$

$v = \frac{d x}{d t} = 2 t C_{0} + 2 C (t - 2)$

$a = \frac{d v}{d t} = 2 C_{0} + 2 C$