A car with bad shocks has a mass of 1500 kg. Before you go for a drive with three of your friends you notice that the car sinks a distance of 6.0 cm when all four of you get in the car. You estimate that the four of you together have a mass of 271 kg. As you are driving down the highway at 65 mph you notice that the car is starting to bounce up and down with large amplitude. You realize that there is a periodic series of small bumps and dips in the road that is driving the bouncing. What is the distance (in meters, to two signi�cant �gures) between adjacent bumps on the road, assuming that damping by the shocks is negligible?
To solve this problem, we can use the concept of the oscillation of the car due to the bumps in the road. The distance between adjacent bumps on the road can be determined using the information provided.
Let's analyze the situation step by step:
1. Calculate the total mass of the car with the four passengers:
- The mass of the car is given as 1500 kg.
- The estimated mass of the four passengers is 271 kg.
- So, the total mass of the car with passengers is 1500 kg + 271 kg = 1771 kg.
2. Determine the effective spring constant of the car's shocks:
- When the car sinks due to the weight of the passengers, it acts like a spring system.
- The distance the car sinks is given as 6.0 cm, which is 0.06 meters.
- The effective spring constant of the shocks can be determined using Hooke's law: F = kx, where F is the force, k is the spring constant, and x is the displacement.
- The force on the car is the weight of the car with passengers, which is given by F = mg, where m is the mass and g is the acceleration due to gravity.
- Plugging in the values, the force F = (1771 kg) * (9.8 m/s^2) = 17350.8 N.
- Setting this force equal to kx, we can solve for the spring constant k: k = F / x = 17350.8 N / 0.06 m = 289180 N/m.
3. Determine the angular frequency of the car's oscillation:
- The car's oscillation can be modeled as a simple harmonic motion.
- The angular frequency ω can be determined using the relation ω = sqrt(k / m), where k is the spring constant and m is the mass.
- Plugging in the values, the angular frequency ω = sqrt(289180 N/m / 1771 kg) = 12.5883 rad/s.
4. Calculate the speed of the car in meters per second:
- The speed of the car is given as 65 mph.
- Converting 65 mph to meters per second, we have 65 mph * (1609.34 m/km) / (3600 s/h) = 29.0576 m/s.
5. Determine the distance between adjacent bumps on the road:
- The distance between adjacent bumps corresponds to the wavelength of the car's oscillation.
- The wavelength λ can be determined using the relation λ = 2πv / ω, where v is the velocity and ω is the angular frequency.
- Plugging in the values, the wavelength λ = (2π * 29.0576 m/s) / (12.5883 rad/s) = 14.4498 m.
Therefore, the distance between adjacent bumps on the road, assuming negligible damping, is approximately 14.45 meters to two significant figures.