The image produced by a concave mirror is located 28 cm in front of the mirror. The focal length of the mirror is 14 cm. How far in front of the mirror is the object located?
_________cm
Use lens equation. Be careful with signs.
To determine the distance of the object from the concave mirror, we can use the mirror equation:
1/f = 1/v - 1/u
Where:
- f is the focal length of the mirror,
- v is the image distance from the mirror, and
- u is the object distance from the mirror.
Given that the focal length (f) is 14 cm and the image distance (v) is 28 cm, we can substitute those values into the equation:
1/14 = 1/28 - 1/u
Simplifying the equation, we get:
1/14 = (1 - 2/u)/28
Next, we can cross-multiply:
u = (28 * 14) / (1 - 2/u)
Multiplying the numerator and denominator of the right side by u, we get:
u = (28 * 14 * u) / (u - 2)
To eliminate the fraction, we can multiply both sides by (u - 2):
u * (u - 2) = (28 * 14 * u)
Expanding the left side, we have:
u^2 - 2u = 392u
Rearranging the equation, we get:
u^2 - 2u - 392u = 0
Combining like terms:
u^2 - 394u = 0
Factoring out the common factor of u:
u(u - 394) = 0
Setting each factor equal to zero:
u = 0 or u - 394 = 0
Since the object distance cannot be zero in this case, we solve for u:
u - 394 = 0
u = 394 cm
Therefore, the object is located 394 cm in front of the concave mirror.