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Vladimir79 [104]
3 years ago
7

A 2 cm tall object is placed 30cm in front of a concave mirror with focal length of 25 cm. What is the height of the image?

Physics
1 answer:
viktelen [127]3 years ago
8 0

Answer:

- 10 cm

Explanation:

h_{o} = height of the object = 2 cm

h_{i} = height of the image

d_{o}  = distance of the object from the mirror = 30 cm  

f  = focal length of the mirror = 25 cm  

d_{i}  = image distance

using the mirror equation

\frac{1}{d_{o}} + \frac{1}{d_{i}} = \frac{1}{f}

\frac{1}{30} + \frac{1}{d_{i}} = \frac{1}{25}

d_{i} = 150 cm

using the equation

\frac{h_{i}}{h_{o}} = \frac{- d_{i}}{d_{o}}

\frac{h_{i}}{2} = \frac{- 150}{30}

h_{i} = - 10 cm

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Consider this 45^{\circ} slope and the trajectory of the skier in a cartesian plane. Since the problem is asking for the displacement vector relative to the point of "lift off", let that particular point be the origin (0, 0).

Assume that the skier is running in the positive x-direction. The line that represents the slope shall point downwards at 45^{\circ} to the x-axis. Since this slope is connected to the ramp, it should also go through the origin. Based on these conditions, this line should be represented as y = -x.

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The question assumes that the skier is in a free-fall motion. In other words, the skier travels with a constant horizontal velocity and accelerates downwards at g (g \approx \rm -9.81\; m\cdot s^{-2} near the surface of the earth.) At t seconds after the skier goes beyond the edge of the ramp, the position of the skier will be:

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  • y-coordinate: \displaystyle -\frac{1}{2}g\cdot t^{2} = -\frac{9.81}{2}\cdot t^{2} meters (constant acceleration with an initial vertical velocity of zero.)

To eliminate t from this expression, solve the equation between t and x for t. That is: express t as a function of x.

x = 30.556\;t\implies \displaystyle t = \frac{x}{30.556}.

Replace the t in the equation of y with this expression:

\begin{aligned} y = &-\frac{9.81}{2}\cdot t^{2}\\ &= -\frac{9.81}{2} \cdot \left(\frac{x}{30.556}\right)^{2}\\&= -0.0052535\;x^{2}\end{aligned}.

Plot the two functions:

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and look for their intersection. Refer to the diagram attached.

Alternatively, equate the two expressions of y (right-hand side of the equation, the part where y is expressed as a function of x.)

-0.0052535\;x^{2} = -x,

\implies x = 190.35.

The value of y can be found by evaluating either equation at this particular x-value: x = 190.35.

y = -190.35.

The position vector of a point (x, y) on a cartesian plane is \displaystyle \left[\begin{array}{l}x \\ y\end{array}\right]. The coordinates of this skier is approximately (190.35, -190.35). The position vector of this skier will be \displaystyle\rm \left[ \begin{array}{c}\rm191\\\rm-191\end{array}\right]. Keep in mind that both numbers in this vectors are in meters.

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