Question

The displacement time graph of two particles P and Q are straight lines inclinced at angles 30 and 60 with time axis the ratio of their velocities is

Answer 1

Solution :

⏭ Given:

✏ The displacement → time graph of two particles P and Q are straight lines inclined at angle 30° and 60° with the time axis.

⏭ To Find:

✏ Ratio of their velocities.

⏭ Concept:

✏ Velocity is defind as ratio of displacement to the time interval.

⏭ Formula:

✏ Formula of velocity is given by

$\star \: \underline{ \boxed{ \bold{ \sf{ \pink{Velocity = \frac{Displacement}{Time}}}}}} \: \star$

✏ For this question, We can find out velocity of particle by....

$\star \: \underline{ \boxed{ \bold{ \sf{ \purple{V = \dfrac{ \triangle{d}}{ \triangle{t}} = Slope \: of \: graph = \red{\tan \theta}}}}}} \: \star$

⏭ Calculation:

$\implies \sf \: \dfrac{V{ \tiny{P}}}{V{ \tiny{Q}}} = \dfrac{ \tan 30 \degree}{ \tan60 \degree} \\ \\ \implies \sf \: \dfrac{V{ \tiny{P}}}{V{ \tiny{Q}}} = \dfrac{ \frac{1}{ \sqrt{3} } }{ \sqrt{3} } = \dfrac{1}{ \sqrt{3} \times \sqrt{3} } \\ \\ \implies \underline{ \boxed{ \bold{ \sf{ \large\orange{ V{ \tiny{P}}: V{ \tiny{Q}} = 1 : 3}}}}} \: \gray{ \bigstar}$

Answer 2

Answer:

• The ratio of their velocities (V_P : V_Q) is 1 : 3

Given:

1. The Angle (θ₁) made by P on time axis is 30°
2. The Angle (θ₂) made by Q on time axis is 60°

Explanation:

$\rule{300}{1.5}$

# Refer the attachment for the figure.

We know,

Velocity is given by the slope of the Displacement - time graph. and we know slope is given by the tan of that angle.

Therefore,

⇒ V_P = tan θ₁

⇒ V_P = tan 30°

⇒ V_P = 1 / √ 3

⇒ V_P = 1 / √ 3

∴ We got the velocity of the Particle P.

$\rule{300}{1.5}$

$\rule{300}{1.5}$

Applying the same concept of slope to the Particle - Q

Therefore,

⇒ V_Q = tan θ₂

⇒ V_Q = tan 60°

⇒ V_Q = √ 3

⇒ V_Q = √ 3

∴ We got the velocity of the Particle Q.

$\rule{300}{1.5}$

$\rule{300}{1.5}$

Now, Applying Ratio,

⇒ V_P / V_Q

⇒ V_P / V_Q = (1 / √ 3) / √ 3

⇒  V_P / V_Q = 1 / (√ 3 ×√ 3)

⇒ V_P / V_Q = 1 / 3

⇒ V_P / V_Q = 1 / 3.

⇒ V_P : V_Q = 1 : 3

⇒ V_P : V_Q = 1 : 3

∴ The ratio of their velocities (V_P : V_Q) is 1 : 3.

$\rule{300}{1.5}$