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?
The correct answer and explanation is :
To solve this problem, we can use the equations of motion for both the Porsche and the Honda, considering their initial conditions and acceleration. We’ll calculate the time it takes for each car to travel the 400 meters and determine who wins the race and by how many seconds.
Step 1: Write the equation of motion
The basic kinematic equation that relates distance, acceleration, and time is:
[
d = \frac{1}{2} a t^2
]
where:
- (d) is the distance covered,
- (a) is the acceleration,
- (t) is the time taken.
Step 2: Porsche’s race time
The Porsche accelerates at (3.5 \, \text{m/s}^2) and starts at (t = 0). We can rearrange the equation to solve for time (t) when the Porsche covers a distance of 400 meters:
[
400 = \frac{1}{2} \times 3.5 \times t^2
]
Solving for (t):
[
t^2 = \frac{400 \times 2}{3.5} = 228.57
]
[
t = \sqrt{228.57} \approx 15.1 \, \text{seconds}
]
So, the Porsche takes approximately 15.1 seconds to complete the 400 meters.
Step 3: Honda’s race time
The Honda gets a 1.0-second head start, so its total time is (t_H = t – 1.0). The Honda accelerates at (3.0 \, \text{m/s}^2), and we can apply the same formula for the distance covered:
[
400 = \frac{1}{2} \times 3.0 \times t_H^2
]
Solving for (t_H):
[
t_H^2 = \frac{400 \times 2}{3.0} = 266.67
]
[
t_H = \sqrt{266.67} \approx 16.3 \, \text{seconds}
]
So, the Honda takes approximately 16.3 seconds to complete the race, considering the head start.
Step 4: Determine who wins
The Porsche takes 15.1 seconds, while the Honda takes 16.3 seconds (considering the head start). Therefore, the Porsche wins the race by:
[
16.3 – 15.1 = 1.2 \, \text{seconds}
]
Conclusion:
The Porsche wins the race by 1.2 seconds. Despite the Honda getting a 1.0-second head start, the Porsche’s higher acceleration allows it to catch up and finish faster.