Problem 54 - Algebra 1: Concepts and Skills

Question

Scientists estimate that a peregrine falcon can dive for its prey at a rate of about 300 feet per second. What is a peregrine falcon’s change in position after diving for 2 seconds?

Step-by-Step Solution

Answer

The peregrine falcon’s change in position after diving for 2 seconds is 600 feet.

Step 1: Understanding the problem

A peregrine falcon is diving at a constant speed of 300 feet per second. Then, it continues diving for 2 seconds. The problem here is to find out how far the falcon will have dived in those 2 seconds.

Step 2: Use formula for distance

As the speed is constant, we can use the formula for covering distance under constant speed, which is Distance = Speed x Time.

Step 3: Calculation

By using the given values, Distance = 300 feet/second x 2 seconds. Performing the multiplication gives Distance = 600 feet. The falcon will have moved by 600 feet in 2 seconds.

Key Concepts

Speed Distance Time Formula

Understanding the relationship between speed, distance, and time is crucial in solving problems within the realm of linear motion. The formula that encapsulates this relationship is surprisingly straightforward: \[ \text{Distance} = \text{Speed} \times \text{Time} \]. This equation helps to calculate how far an object has traveled given its speed and the amount of time it has been moving.

In the context of our peregrine falcon, we apply this formula to calculate its change in position after diving for a set period. If a falcon dives at a constant speed of 300 feet per second, and this action continues for 2 seconds, we calculate the distance by multiplying the speed by the time (\(300 \times 2\)). This makes the complex task of distance calculation simple and manageable, more so when the speed remains unchanged throughout the motion.

Algebraic Problem-Solving

Algebraic problem-solving is a systematic approach that requires us to understand the problem, identify what we know, what we need to find out, and the relationship between these quantities. In this case, we know the speed of the falcon and the time duration of its dive. Our objective is to find out the change in position or distance covered during this time.

We use algebraic expressions to represent the problem's parameters. For example, speed (S) can be denoted as 300 feet/second, time (T) as 2 seconds, and distance (D) as the unknown we are trying to find. By substituting the known values into the speed-distance-time formula \(D = S \times T\), we transform a word problem into an algebraic equation that is solvable through basic arithmetic operations.

Linear Motion

Linear motion refers to the movement of an object along a straight line. In classical mechanics, it is assumed that an object in linear motion will either remain at rest or move at a constant speed unless acted on by a force. This forms the basis of problems involving constant speed, like the dive of a peregrine falcon.

Understanding linear motion allows us to predict and account for the position of an object at any given time. When a falcon dives at a steady rate of 300 feet per second, we anticipate that its position changes uniformly over time. Thus, the distance covered after 2 seconds can be precisely calculated using the formula previously mentioned. This situation perfectly exemplifies the linear motion concept where the direction of travel remains unchanged and the velocity is constant.