Question

The driver sitting inside a parked car is watching vehicles approaching from behind with the help of his side view mirror, which is a convex mirror with radius of curvature \( R = 2 \, \text{m} \). Another car approaches him from behind with a uniform speed of 90 km/hr. When the car is at a distance of 24 m from him, the magnitude of the acceleration of the image of the side view mirror is \( a \). The value of \( 100a \) is _____________ m/s\(^2\).

Hint:

To find the acceleration of an image in a moving mirror, use the formula for magnification and differentiate with respect to time.

Answer 1

Correct Answer: 8

\[ v = \frac{uf}{u-f} = \frac{-24 \cdot 1}{-24 - 1} = \frac{24}{25} \] Explanation: This calculates the image distance $v$ using the lens formula, given the object distance $u$ and focal length $f$. \[ m = \frac{-v}{u} = -\frac{24}{25(-24)} = \frac{1}{25} \] Explanation: This calculates the magnification $m$ using the formula $m = -v/u$. \[ \frac{1}{v} - \frac{1}{u} = \frac{1}{f} \] Explanation: This is the lens formula relating object distance $u$, image distance $v$, and focal length $f$. \[ v_1 = -m^2 \cdot v_0 = -\frac{1}{(25)^2} \cdot 25 = -\frac{1}{25} \] Explanation: This calculates the velocity of the image $v_1$ given the magnification $m$ and the velocity of the object $v_0$. \[ \text{Diff.} \quad \frac{-1}{v^2} \left( \frac{dv}{dt} \right) + \frac{1}{u^2} \left( \frac{du}{dt} \right) = 0 \quad \left[ \frac{dv}{dt} = v_1; \frac{du}{dt} = v_0 \right] \] Explanation: This is the differentiated form of the lens formula, expressing the relationship between the velocities of the object and image. \[ \frac{+2}{v^3} \cdot (v_1)^2 - \frac{1}{v^2} \cdot a_1 - \frac{2}{u^3} \cdot (v_0)^2 + \frac{1}{u^2} \cdot a_0 = 0 \] Explanation: This is the second derivative of the lens formula, relating the accelerations of the object and image. \[ \frac{a_1}{v^2} = \frac{2}{v^3} \cdot v_1^2 - \frac{2}{u^3} \cdot v_0^2 \] Explanation: This rearranges the previous equation to solve for the acceleration of the image $a_1$. \[ a_1 = \frac{2}{v} \cdot v_1^2 - \frac{2v^2}{u^3} \cdot v_0^2 \] Explanation: This isolates $a_1$ by multiplying both sides by $v^2$. \[ = \frac{2 \cdot 25}{24} \cdot \frac{1}{25} \cdot \frac{1}{25} - \frac{2}{(24)^3} \cdot \frac{24}{25} \cdot \frac{24}{25} \cdot 25 \] Explanation: This substitutes the calculated values of $v$, $v_1$, $u$, and $v_0$ into the expression for $a_1$. \[ a_1 = \frac{2}{24 \cdot 25} - \frac{2}{24} \] Explanation: This simplifies the expression by canceling out common terms. \[ a_1 = \frac{2}{24} \cdot \frac{-24}{25} = -\frac{2}{25} \] Explanation: This further simplifies the expression to find the value of $a_1$. \[ 100 a_1 = \frac{2}{25} \times 100 = 8 \] Explanation: This calculates $100a_1$ by multiplying $a_1$ by 100.