Mathematics

Mathematics

Problem

An observer stands 80 m80\text{ m} from a straight road. A car travels along the road at a constant speed of 25 m/s25\text{ m/s}. Let θ\theta be the angle between the observer’s line of sight and the perpendicular from the observer to the road. What is the magnitude of d2θdt2\dfrac{d^{2}\theta}{dt^{2}} (in rad/s2^{2}) when the car is 100 m100\text{ m} from the point on the road closest to the observer? a) 0.0740.074 b) 0.0180.018 c) 0.0370.037 d) 0.0280.028
Intuitive explanation

This is a related rates problem involving second derivatives—you need to find how fast the rate of change of the angle is itself changing. Start by setting up a right triangle where the observer's perpendicular distance (80 m) is fixed, the car's position along the road varies with time, and θ\theta is the angle at the observer. Use tan⁡(θ)\tan(\theta) to relate position to angle, then differentiate twice with respect to time to get d2θdt2\frac{d^2\theta}{dt^2}. The key insight is that even though the car moves at constant speed, the angle changes at a non-constant rate—the second derivative captures this acceleration of the angle itself.

Intuitive explanation

This is a related rates problem involving second derivatives—you need to find how fast the rate of change of the angle is itself changing. Start by setting up a right triangle where the observer's perpendicular distance (80 m) is fixed, the car's position along the road varies with time, and θ\theta is the angle at the observer. Use tan⁡(θ)\tan(\theta) to relate position to angle, then differentiate twice with respect to time to get d2θdt2\frac{d^2\theta}{dt^2}. The key insight is that even though the car moves at constant speed, the angle changes at a non-constant rate—the second derivative captures this acceleration of the angle itself.

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