A Porsche challenges a Honda to a 400 m race. Because the Porsche's acceleration of 3.5 m/s2 is larger than the Honda's 3.0 m/s2, the Honda gets a 1.0 s head start. Who wins? By how many seconds?

Ch 02: Kinematics in One Dimension

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 02: Kinematics in One Dimension

Problem 18

Chapter 2, Problem 18

A Porsche challenges a Honda to a 400 m race. Because the Porsche's acceleration of 3.5 m/s² is larger than the Honda's 3.0 m/s², the Honda gets a 1.0 s head start. Who wins? By how many seconds?

Verified step by step guidance

Step 1: Write the equation for the displacement of each vehicle using the kinematic formula for uniformly accelerated motion: \( x = x_0 + v_0 t + \frac{1}{2} a t^2 \). For simplicity, assume \( x_0 = 0 \) and \( v_0 = 0 \) for both vehicles.

Step 2: For the Honda (red bike), account for the 1.0 s head start. Its displacement equation becomes \( x_{Honda} = \frac{1}{2} a_{Honda} (t + 1)^2 \), where \( a_{Honda} = 3.0 \, \text{m/s}^2 \).

Step 3: For the Porsche (black bike), its displacement equation is \( x_{Porsche} = \frac{1}{2} a_{Porsche} t^2 \), where \( a_{Porsche} = 3.5 \, \text{m/s}^2 \).

Step 4: Set both displacement equations equal to 400 m (the race distance) and solve for the time \( t \) for each vehicle. For the Honda: \( 400 = \frac{1}{2} (3.0) (t + 1)^2 \). For the Porsche: \( 400 = \frac{1}{2} (3.5) t^2 \).

Step 5: Compare the times \( t \) for each vehicle to determine who finishes the race first. Subtract the smaller time from the larger time to find the time difference by which the winner beats the other.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Acceleration

Acceleration is the rate of change of velocity of an object with respect to time. It is a vector quantity, meaning it has both magnitude and direction. In this scenario, the Porsche has an acceleration of 3.5 m/s², while the Honda has 3.0 m/s². This difference in acceleration will affect how quickly each vehicle can reach the finish line after the head start.

Kinematics Equations

Kinematics equations describe the motion of objects under constant acceleration. They relate displacement, initial velocity, final velocity, acceleration, and time. For this race, we can use these equations to calculate the distance each bike travels over time, taking into account the Honda's 1-second head start and their respective accelerations.

Relative Motion

Relative motion refers to the calculation of the motion of an object as observed from a particular reference point, which can be stationary or moving. In this race, the Honda's head start creates a relative position advantage, and understanding how this affects the race outcome is crucial. We need to analyze how the Porsche's acceleration allows it to catch up and potentially overtake the Honda.

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Intro to Relative Motion (Relative Velocity)

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