Question

A powerful 0.54-W laser emitting 670-nm photons shines on a black sail of a tiny 0.10-g cart that can coast on a frictionless track.

Part A

Determine the force of the light on the sail. Assume that the light is totally absorbed by the sail.

Express your answer to two significant figures and include the appropriate units.

F =	7.01•106N

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Part B

What time interval is needed for the cart's speed to increase from zero to 2.0 m/s?

Express your answer using two significant figures.

t =	.37•10−3	s

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Answer 1

Part A

The number of photons per unit time emitted by the laser is

Here, power of the laser light is P, wavelength of the photon is , Planck’s constant is h and speed of light in vacuum is c.

Substitute 0.54 W for P, 670 nm for , 6.63 x 10^-34 for h and 3 x 10⁸ m/s for c in the above equation,

From de Broglie relation, the momentum of the photon is

Now, the force of the light on the sail is equal to the number of photon per unit second times momentum of the photon

F = np

    =(18.2 x 10¹⁷ /s) (9.89 x 10^-28 kg m/s)

    = 1.7999 x 10^-9 N

Rounding off to two significant figures, the force of the light on the snail is 1.8 x 10^-9 N.

Part B:

Let m be the mass of the cart.

Let t be the time required to change the velocity of the cart.

The rate of change of momentum is equal to force times time interval.

It is

mv – mu = Ft

t = m(v-u) /F

= (0.1 g)(1 kg / 1000 g) (2.0 m/s – 0) / 1.8 x 10^-9 N

= 0.111 x 10⁶ s

Rounding off to two significant figures, the time interval needed to increase the speed of cart is 0.11 x 10⁶ s.