Tutor profile: Rahil V.

Find $$dy/dx$$ if $$y=\sqrt{\sin(x)}$$

Let $$u=\sin(x)$$, then we have $$y=\sqrt{u}=u^{1/2}$$. So, $$\frac{dy}{du}$$ $$=\frac{1}{2}u^{-1/2}$$ $$=\frac{1}{2\sqrt{u}}$$. Now,$$\frac{du}{dx}=\cos(x)$$. Now, $$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} $$. Hence, $$\frac{dy}{dx} = \frac{1}{2\sqrt{u}} \times \cos(x)$$ Therefore, $$\frac{dy}{dx} = \frac{\cos(x)}{2\sqrt{\sin(x)}} $$.

If a feather and a hammer are both dropped from the same height on Earth then which one would reach the ground first? What would happen if the same experiment was carried out on Moon?

Lets start with the Moon. Since the Moon does not have any atmosphere, both the feather and the hammer would fall down with the same acceleration which is the gravitational acceleration of the moon. Hence, they will both reach the ground at the same time. This experiment has been done by astronauts who visited the moon (https://www.youtube.com/watch?v=KDp1tiUsZw8&ab_channel=AIRBOYD). On Earth, due to air resistance, the feather will experience a high drag force that was absent on the moon, This will result in the hammer reaching the ground first compared to the feather.

How do airplanes fly and why do racing cars have spoilers?

An airplane flies due to an upward force, called lift, generated by the wings. If one were to cut an airplane wing and look at the cross section then one would see the shape of an airfoil. When air flows around the airfoil, the air on the upper surface flow faster compared to the air on the bottom surface. This creates lower pressure on the upper surface compared to the bottom surface. This pressure difference gives rise to a force that acts from high pressure to low pressure i.e. pointing in the upward direction. This is the lift force that makes an airplane fly. The cross section of the spoiler of a racing car is an upside down airfoil. So when air flows over it, now the bottom surface has lower pressure compared to the top surface giving rise to a downward force. This downward force helps keep the racing car stuck to the ground when the car is turning at high speeds.