Question

At t = 0.0 s, a truck is traveling east at a constant speed of s = 87.3 km/h. At an intersection d = 37.3 km ahead, a car is traveling north at constant speed of v = 53.1 km/h.

1. What is the expression for the distance r between the truck and the car as a function of time. Use variables from the problem statement for your equation.
2. What is the expression for the time at which the distance between the car and the truck has it's minimum value. Use variable from the problem statement for your equation.

Answer 1

To determine the distance between a truck and a car traveling at right angles to each other at constant speeds, the Pythagorean theorem is applied with respect to time, while calculus is used to find the time when this distance is at its minimum.

Step-by-step explanation:

To find the expression for the distance r between the truck and the car as a function of time, we can use the Pythagorean theorem. The truck is traveling east at a constant speed of s = 87.3 km/h and the car is traveling north at a constant speed of v = 53.1 km/h, and initially, they are d = 37.3 km apart.

Let t be the time in hours after t = 0.0 s.

The truck travels a distance of s*t km to the east.

The car travels a distance of v*t km to the north.

The distance r between the truck and the car at any time t can be found using the Pythagorean theorem: r = √((d + s*t)2 + (v*t)2).

To find the expression for the time at which the distance between the truck and the car is at its minimum, we would differentiate the equation for r with respect to time t and set the derivative to zero to solve for t. This involves calculus, which may be beyond the scope of high school physics, but the process would involve setting the derivative of the sqrt((d + s*t)2 + (v*t)2) with respect to t to zero and solving for t.

Answer 2

To find the distance between the truck and car as a function of time, the Pythagorean theorem can be used with the displacements of the vehicles, resulting in a square root function. The time at which the distance is minimized can be found by differentiating this function and solving for t.

Step-by-step explanation:

The calculation involves physics concepts pertaining to relative motion and kinematics. To find the expression for the distance r between the truck and the car as a function of time, we use the Pythagorean theorem. The truck's displacement from the starting point after time t is st, while the car's displacement is vt. To find the distance r between them at any time t, we have:

r(t) = √ [(st)^2 + (vt + d)^2]

To determine the time at which this distance is minimized, we differentiate this function with respect to t and set the derivative equal to zero. Solving this for t will give us the time at which r is minimized.