Problem 45 - College Physics

Question

Two bicyclists start a sprint from rest, each riding with a constant acceleration. Bicyclist \(A\) has twice the acceleration of bicyclist \(B ;\) however, bicyclist \(B\) rides for twice as long as bicyclist \(A\). What is the ratio of the distance traveled by bicyclist \(A\) to that traveled by bicyclist \(B\) ? What is the ratio of the speed of bicyclist \(A\) to that of bicyclist \(B\) at the end of their sprint?

Step-by-Step Solution

Answer

Distance ratio \( A:B \) is \( 1:2 \), Speed ratio \( A:B \) is \( 1:1 \).

Step 1: Set Up the Problem

Let the acceleration of bicyclist \( B \) be \( a \). Therefore, the acceleration of bicyclist \( A \), being twice that, is \( 2a \). Let the time for bicyclist \( A \) be \( t \), and accordingly, the time for bicyclist \( B \) is \( 2t \). We need to find two ratios: one for distances and one for speeds.

Step 2: Calculate Distance for Bicyclist A

Using the formula for distance with constant acceleration, \( s = \frac{1}{2} a t^2 \), substitute the values for bicyclist \( A \):\[ s_A = \frac{1}{2} \times 2a \times t^2 = a t^2\].

Step 3: Calculate Distance for Bicyclist B

Using the same formula, \( s = \frac{1}{2} a t^2 \), substitute the values for bicyclist \( B \):\[ s_B = \frac{1}{2} \times a \times (2t)^2 = 2a t^2\].

Step 4: Find the Ratio of Distances

To find the ratio of the distances traveled by bicyclists \( A \) and \( B \), divide their distances:\[ \text{Distance Ratio} = \frac{s_A}{s_B} = \frac{a t^2}{2a t^2} = \frac{1}{2}\].

Step 5: Calculate Speed for Bicyclist A

Using the formula for speed with constant acceleration, \( v = at \), for bicyclist \( A \):\[ v_A = 2a \times t = 2at\].

Step 6: Calculate Speed for Bicyclist B

Similarly, use the formula \( v = at \), for bicyclist \( B \):\[ v_B = a \times 2t = 2at\].

Step 7: Find the Ratio of Speeds

To find the ratio of the speeds of bicyclists \( A \) and \( B \), divide their speeds:\[ \text{Speed Ratio} = \frac{v_A}{v_B} = \frac{2at}{2at} = 1\].

Key Concepts

Constant Acceleration

Acceleration is a key concept in kinematics, describing the change in velocity over time. When acceleration is constant, it means the speed of an object increases at a steady rate. In our exercise, both bicyclists start from rest, which means their initial speed is zero. This is a crucial condition because any changes in distance and speed during their ride solely depend on their constant acceleration. For bicyclist A, the acceleration can be represented as \( 2a \) because it is twice the acceleration of bicyclist B, whose acceleration is \( a \). Constant acceleration is advantageous for calculations, as it lets you predict the speed and distance covered using straightforward equations, such as \( v = at \) for speed and \( s = \frac{1}{2} a t^2 \) for distance.

Distance Traveled

Distance is defined as the total path covered by an object and can be directly computed when dealing with constant acceleration. For any object under uniform acceleration starting from rest, the distance traveled can be calculated using the formula:\[s = \frac{1}{2} a t^2\]- Bicyclist A has an acceleration of \( 2a \) and rides for time \( t \), covering a distance \( s_A = a t^2 \).- Bicyclist B, with acceleration \( a \), rides for a longer time, \( 2t \), so the distance \( s_B = 2a t^2 \).By comparing these distances, we learn the ratio of their distances traveled:\[\frac{s_A}{s_B} = \frac{a t^2}{2a t^2} = \frac{1}{2}\]Thus, bicyclist A covers half the distance that bicyclist B travels in the same time.

Speed Ratio

Speed represents the rate at which an object covers distance. With constant acceleration, speed at any time can be determined using the equation:\[v = at\]- For bicyclist A, who has twice the acceleration \( 2a \), the speed is \( v_A = 2at \) at time \( t \).- Bicyclist B travels with acceleration \( a \) for time \( 2t \), also reaching a speed \( v_B = 2at \).Interestingly, both bicyclists reach the same final speed, since although bicyclist A accelerates faster, bicyclist B compensates by riding for twice as long. Consequently, this gives them a speed ratio:\[\frac{v_A}{v_B} = 1\]Meaning at the end of the sprint, both bicyclists reach the same speed.

Time of Motion

The time of motion indicates how long an object continues to move. It significantly affects both the distance and speed when dealing with constant acceleration.In this scenario, bicyclist A moves for time \( t \), while bicyclist B continues for \( 2t \). The difference in their motion duration directly influences their distance traveled but not their final speed.- Bicyclist A has a shorter time to exert its higher acceleration.- Conversely, bicyclist B, despite having a lower acceleration, compensates with a longer timeframe of movement.Time is critical because it allows bicyclist B to continue gaining speed and cover more ground, ultimately impacting the ratio of their distances. However, since their speeds end up the same, the impact of time on speed equalizes between them. Understanding time's role can clarify why objects with different accelerations and motion durations achieve unique results under constant acceleration.