NEET Previous Year Questions - Motion in A Straight Line

Q 1 :
In some appropriate units, time $(t)$ and position $(x)$ relation of a moving particle is given by $t = x^{2} + x .$ The acceleration of the particle is: [2025]

$+ \frac{2}{{(x + 1)}^{3}}$
$+ \frac{2}{2 x + 1}$
$- \frac{2}{{(x + 2)}^{3}}$
$- \frac{2}{{(2 x + 1)}^{3}}$

1

2

3

4

(4)

Given, $t = x^{2} + x$

Differentiating with respect to 't' on both sides,

$\Rightarrow \frac{d t}{d t} = 2 x \frac{d x}{d t} + \frac{d x}{d t}$

$\Rightarrow 1 = 2 x v + v$ ...(i)

$\Rightarrow 1 = v (2 x + 1)$

$v = \frac{1}{2 x + 1}$ ...(ii)

Again, differentiating equation (i) with respect to 't',

$0 = 2 (\frac{d x}{d t} . v + x \frac{d v}{d t}) + \frac{d v}{d t}$

$0 = 2 (v^{2} + x . a) + a$

$0 = 2 v^{2} + 2 x a + a$

$0 = [2 v^{2} + a (2 x + 1)]$

$\Rightarrow 2 v^{2} = - a (2 x + 1)$

Using equation (ii), we get

$2 {(\frac{1}{2 x + 1})}^{2} = - a (2 x + 1)$

$∴$ $a = \frac{- 2}{{(2 x + 1)}^{3}}$

Q 2 :
A particle is moving along x-axis with its position $(x)$ varying with time $(t)$ as $x = α t^{4} + β t^{2} + γ t + δ .$ The ratio of its initial acceleration, respectively, is [2024]

$2 α : δ$
$γ : 2 δ$
$4 α : β$
$γ : 2 β$

1

2

3

4

(4)

Given, position, $(x) = α t^{4} + β t^{2} + γ t + δ$

Velocity, $(v) = \frac{d x}{d t} = 4 α t^{2} + 2 β t + γ$

Initial velocity, $v$ $(a t t = 0) = γ$ ...(i)

Acceleration, $(a) = \frac{d v}{d t} = 12 α t^{2} + 2 β$

Initial acceleration, $a$ $(a t t = 0) = 2 β$ ...(ii)

From equation (i) and (ii)

$\frac{v (0)}{a (0)} = \frac{γ}{2 β} = γ : 2 β$

Q 3 :
The velocity ( $v$ ) – time ( $t$ ) plot of the motion of a body is shown below.

The acceleration ( $a$ ) – time ( $t$ ) graph that best suits this motion is [2024]

1

2

3

4

(3)

Slope of velocity-time graph gives acceleration.

AB: Slope is positive and constant, so acceleration is positive and constant.
BC: Slope is zero, so acceleration is zero.
CD: Slope is negative and constant, so acceleration is negative and constant.

Q 4 :
A particle of unit mass undergoes one-dimensional motion such that its velocity varies according to $v (x) = β x^{- 2 n},$ where $β$ and $n$ are constants and $x$ is the position of the particle. The acceleration of the particle as a function of $x$ , is given by [2015]

$- 2 β^{2} x^{- 2 n + 1}$
$- 2 n β^{2} e^{- 4 n + 1}$
$- 2 n β^{2} x^{- 2 n - 1}$
$- 2 n β^{2} x^{- 4 n - 1}$

1

2

3

4

(4)

According to question, velocity of unit mass varies as

$v (x) = β x^{- 2 n}$ ...(i), $\frac{d v}{d x} = - 2 n β x^{- 2 n - 1}$ ...(ii)

Acceleration of the particle is given by

$a = \frac{d v}{d t} = \frac{d v}{d x} \times \frac{d x}{d t} = \frac{d v}{d x} \times v$

Using equation (i) and (ii), we get

$a = (- 2 n β x^{- 2 n - 1}) \times (β x^{- 2 n}) = - 2 n β^{2} x^{- 4 n - 1}$

Topic Question Set

Q 1 :
In some appropriate units, time $(t)$ and position $(x)$ relation of a moving particle is given by $t = x^{2} + x .$ The acceleration of the particle is: [2025]

Q 2 :
A particle is moving along x-axis with its position $(x)$ varying with time $(t)$ as $x = α t^{4} + β t^{2} + γ t + δ .$ The ratio of its initial acceleration, respectively, is [2024]

Q 3 :
The velocity ( $v$ ) – time ( $t$ ) plot of the motion of a body is shown below.

The acceleration ( $a$ ) – time ( $t$ ) graph that best suits this motion is [2024]

Q 4 :
A particle of unit mass undergoes one-dimensional motion such that its velocity varies according to $v (x) = β x^{- 2 n},$ where $β$ and $n$ are constants and $x$ is the position of the particle. The acceleration of the particle as a function of $x$ , is given by [2015]

Want Us to Email you About Special Offers & Updates?

Topic Question Set

Q 1 : In some appropriate units, time (t) and position (x) relation of a moving particle is given by t=x2+x. The acceleration of the particle is: [2025]

Q 2 : A particle is moving along x-axis with its position (x) varying with time (t) as x=αt4+βt2+γt+δ. The ratio of its initial acceleration, respectively, is [2024]

Q 3 : The velocity (v) – time (t) plot of the motion of a body is shown below. The acceleration (a) – time (t) graph that best suits this motion is [2024]

Q 4 : A particle of unit mass undergoes one-dimensional motion such that its velocity varies according to v(x)=βx-2n, where β and n are constants and x is the position of the particle. The acceleration of the particle as a function of x, is given by [2015]

Want Us to Email you About Special Offers & Updates?

Q 1 :
In some appropriate units, time $(t)$ and position $(x)$ relation of a moving particle is given by $t = x^{2} + x .$ The acceleration of the particle is: [2025]

Q 2 :
A particle is moving along x-axis with its position $(x)$ varying with time $(t)$ as $x = α t^{4} + β t^{2} + γ t + δ .$ The ratio of its initial acceleration, respectively, is [2024]

Q 3 :
The velocity ( $v$ ) – time ( $t$ ) plot of the motion of a body is shown below.

The acceleration ( $a$ ) – time ( $t$ ) graph that best suits this motion is [2024]

Q 4 :
A particle of unit mass undergoes one-dimensional motion such that its velocity varies according to $v (x) = β x^{- 2 n},$ where $β$ and $n$ are constants and $x$ is the position of the particle. The acceleration of the particle as a function of $x$ , is given by [2015]