A ball is thrown vertically upward with an initial velocity of 150 m/s. The ratio of velocity after 3 s and 5 s is x+1x. The value of x is: Take g = 10 m/s^2.

Given: Initial velocity u = 150 m/s and acceleration due to gravity g = 10 m/s^2 acting downward. Upward direction is taken as positive. Find: The value of x if the ratio of velocities after 3 s and 5 s is x+1x. Using the kinematic relation: v = u - gt At t = 3 s, v1 = 150 - 10(3) = 120 m/s At t = 5 s, v2 = 150 - 10(5) = 100 m/s Therefore, v1v2 = 120100 = 65 Given that x+1x = 65 So, 5(x+1) = 6x 5x + 5 = 6x x = 5 Therefore, the correct option is B. Given: u = 150 m/s and g = 10 m/s^2. Find: The value of x. The solution gives the ratio directly as: v3v5 = u-g 3u-g 5 Substituting values, v3v5 = 150-30150-50 = 120100 = 65 Hence, x+1x = 65 Comparing numerator and denominator gives x = 5. The correct option is B.

A ball is thrown vertically upward with an initial velocity of 15… — Solution | JEE Physics

MCQEasyJEE 2023Motion in a Straight Line

A ball is thrown vertically upward with an initial velocity of $150 \, \text{m/s}$ . The ratio of velocity after $3 \, \text{s}$ and $5 \, \text{s}$ is $\frac{x+1}{x}$ . The value of $x$ is:

Take $g = 10 \, \text{m/s}^2$ .

A
$6$
B
$5$
C
$-5$
D
$10$

Answer

Correct answer:B

Step-by-step solution

Standard Method

Given: Initial velocity $u = 150 \, \text{m/s}$ and acceleration due to gravity $g = 10 \, \text{m/s}^2$ acting downward. Upward direction is taken as positive.

Find: The value of $x$ if the ratio of velocities after $3 \, \text{s}$ and $5 \, \text{s}$ is $\frac{x+1}{x}$ .

Using the kinematic relation:

v = u - gt

At $t = 3 \, \text{s}$ ,

v_1 = 150 - 10(3) = 120 \, \text{m/s}

At $t = 5 \, \text{s}$ ,

v_2 = 150 - 10(5) = 100 \, \text{m/s}

Therefore,

\frac{v_1}{v_2} = \frac{120}{100} = \frac{6}{5}

Given that

\frac{x+1}{x} = \frac{6}{5}

So,

5(x+1) = 6x

5x + 5 = 6x

x = 5

Therefore, the correct option is B.

Using the ratio directly

Given: $u = 150 \, \text{m/s}$ and $g = 10 \, \text{m/s}^2$ .

Find: The value of $x$ .

The solution gives the ratio directly as:

\frac{v_3}{v_5} = \frac{u-g\times3}{u-g\times5}

Substituting values,

\frac{v_3}{v_5} = \frac{150-30}{150-50} = \frac{120}{100} = \frac{6}{5}

Hence,

\frac{x+1}{x} = \frac{6}{5}

Comparing numerator and denominator gives $x = 5$ . The correct option is B.

Common mistakes

Using $v = u + gt$ with $g = 10 \, \text{m/s}^2$ while also taking upward as positive is incorrect. Gravity acts downward, so the acceleration must be negative in this sign convention. Use $v = u - gt$ .
Treating the ratio $\frac{6}{5}$ as the final answer is incorrect because the question asks for the value of $x$ , not the ratio itself. After finding the ratio, equate $\frac{x+1}{x} = \frac{6}{5}$ and solve for $x$ .
Confusing speed and velocity signs can lead to errors. In this case, both velocities at $3 \, \text{s}$ and $5 \, \text{s}$ are still upward and positive, so the ratio remains positive. Always check the direction before forming the ratio.

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