Question

Based on Figure \(3,\) if \(c\) represents the speed of light, how long did it take each radar pulse to make the round-trip between the transceiver and the ball? F. 2\(\frac{D}{c}\) G. \(\frac{R}{c}\) H. 2\(\frac{c}{D}\) J. \(\frac{c}{R}\)

Short answer

A: The formula for the time taken is t = 2\(\frac{D}{c}\).

Step-by-step solution

Calculate the round-trip distance

Since the radar pulse makes a round-trip between the transceiver and the ball, the total distance traveled by the pulse is two times the actual distance between the transceiver and the ball. Let the actual distance between the two points be denoted by 'D'. Hence, the round-trip distance will be 2D.

Use the time formula with the speed of light

To find the time taken, we use the formula time (t) = distance / speed. In this case, the speed is the speed of light, represented by 'c'. So, we will calculate the time t by dividing the round-trip distance (2D) by the speed of light (c). t = (2D) / c

Choose the correct answer

Comparing our result with the given options, we see that the correct answer is: F. 2\(\frac{D}{c}\)

Key concepts

Speed of Light

The speed of light, denoted as 'c', is a fundamental constant in physics, representing the speed at which light travels in a vacuum. The exact value is approximately 299,792 kilometers per second (or about 186,282 miles per second). Understanding this speed is crucial when studying phenomena involving light, such as reflection, refraction, and in this case, radar detection.

When calculating how long it takes for light to travel a certain distance, knowing that 'c' is constant can simplify the process significantly. In the context of ACT Math Prep, comprehending that nothing can travel faster than the speed of light helps to emphasize the invariant nature of 'c' in equations involving time and distance.

Distance Formula

In physics and mathematics, the distance formula is used to determine the distance between two points. Traditionally, when applied to a coordinate plane, it takes the form of \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \), where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points.

However, in a more straightforward scenario such as the one given in the exercise, where light travels directly between two points and back, we can simplify the calculation. The total distance covered during a round-trip is twice the one-way distance—denoted as 2D. It's essential for students to grasp this concept as it is commonly used in problems involving motion, especially in cases where an object or wave returns to its starting point.

Time Calculation

Time calculation in physics often involves finding how long it takes for an object or wave to traverse a particular distance at a given speed. The basic formula is \( t = \frac{d}{s} \), which means time equals distance divided by speed.

In the context of the given problem, we are calculating the time it takes for a radar pulse to travel back and forth between a transceiver and a ball. Understanding that time is the dependent variable in this scenario is pivotal. For the pulse making a round trip, the distance 'd' is represented as '2D' (twice the one-way distance), and the speed 's' is the speed of light 'c'. The ACT Math Prep exercise showcases the direct application of this formula, reinforcing the relationship between time, distance, and speed in a straightforward and real-world applicable scenario.