(D) The position of bus $P$ is given by $X_{P}(t) = \alpha t + \beta t^{2}$.The velocity of bus $P$ is $V_{P}(t) = \frac{dX_{P}}{dt} = \alpha + 2\beta

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(D) The position of bus $P$ is given by $X_{P}(t) = \alpha t + \beta t^{2}$.The velocity of bus $P$ is $V_{P}(t) = \frac{dX_{P}}{dt} = \alpha + 2\beta t$.The position of bus $Q$ is given by $X_{Q}(t) = ft - t^{2}$.The velocity of bus $Q$ is $V_{Q}(t) = \frac{dX_{Q}}{dt} = f - 2t$.For the buses to have the same velocity,we set $V_{P}(t) = V_{Q}(t)$:$\alpha + 2\beta t = f - 2t$.Rearranging the terms to solve for $t$:$2\beta t + 2t = f - \alpha$.$t(2\beta + 2) = f - \alpha$.$t = \frac{f - \alpha}{2(1 + \beta)}$.

Class 11 Physics Motion in a Straight Line Mix Examples-Motion in Straight Line

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Two buses $P$ and $Q$ start from a point at the same time and move in a straight line and their positions are represented by $X_{P}(t) = \alpha t + \beta t^{2}$ and $X_{Q}(t) = ft - t^{2}$. At what time,do both the buses have the same velocity?

A
$\frac{\alpha-f}{1+\beta}$
B
$\frac{\alpha+f}{2(\beta-1)}$
C
$\frac{\alpha+f}{2(1+\beta)}$
D
$\frac{f-\alpha}{2(1+\beta)}$

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Motion in a Straight Line

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Mix Examples-Motion in Straight Line

If a car covers $2/5^{th}$ of the total distance with speed $v_1$ and $3/5^{th}$ of the total distance with speed $v_2$,then the average speed is:

Medium

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Choose the correct statement.

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$A$ ball is dropped vertically from a height $d$ above the ground. It hits the ground and bounces up vertically to a height $d/2$. Neglecting subsequent motion and air resistance,its velocity $v$ varies with the height $h$ above the ground as:

DifficultIIT 2000

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$A$ man in a car at location $Q$ on a straight highway is moving with speed $v$. He decides to reach a point $P$ in a field at a distance $d$ from the highway (point $M$) as shown in the figure. The speed of the car in the field is half of that on the highway. What should be the distance $RM$ so that the time taken to reach $P$ is minimum?

DifficultJEE MAIN 2018

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The equation of motion of a body is $\frac{dv}{dt} = -4v + 8$,where $v$ is the velocity in $m/s$ and $t$ is the time in seconds. The initial velocity of the particle was zero. Then,

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Two buses $P$ and $Q$ start from a point at the same time and move in a straight line and their positions are represented by $X_{P}(t) = \alpha t + \beta t^{2}$ and $X_{Q}(t) = ft - t^{2}$. At what time,do both the buses have the same velocity?

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If a car covers $2/5^{th}$ of the total distance with speed $v_1$ and $3/5^{th}$ of the total distance with speed $v_2$,then the average speed is:

Choose the correct statement.

$A$ ball is dropped vertically from a height $d$ above the ground. It hits the ground and bounces up vertically to a height $d/2$. Neglecting subsequent motion and air resistance,its velocity $v$ varies with the height $h$ above the ground as:

The equation of motion of a body is $\frac{dv}{dt} = -4v + 8$,where $v$ is the velocity in $m/s$ and $t$ is the time in seconds. The initial velocity of the particle was zero. Then,

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