When faced with a physics problem involving motion, understanding the underlying principles and applying the correct equations is crucial. Let’s analyze the scenario of a car traveling at 30m/s runs out of gas while ascending a 20-degree slope and determine how far it will coast before rolling back down.
The initial attempt at solving this problem highlights some common pitfalls. It’s important to carefully consider the initial conditions, the relevant physics equations, and the reasonableness of the calculated results.
Understanding the Problem
The car is traveling at an initial velocity (Vo) of 30 m/s up a 20-degree slope. Gravity acts on the car, causing it to decelerate. The goal is to find the distance (Y) the car travels up the slope before its final velocity (Vf) becomes 0.
Relevant Equations
The following kinematic equations are relevant to this problem:
- Vf = Vo + at
- Vf^2 = Vi^2 + 2aY
- Y = Yo + Vo(t) + 1/2 aT^2
Where:
- Vf = Final velocity
- Vo = Initial velocity
- a = Acceleration
- t = Time
- Y = Distance
- Yo = Initial Distance (often 0)
Identifying the Error in the Initial Attempt
The initial attempt incorrectly uses 30sin20 as the initial speed. The initial speed of 30m/s is up the slope. The component of gravity acting down the slope causes the deceleration. The acceleration is calculated using the component of gravity acting along the slope, which is -g*sin(θ), where g is the acceleration due to gravity (9.8 m/s²) and θ is the angle of the slope (20 degrees).
Correct Solution
-
Calculate the acceleration:
a = -9.8 * sin(20) = -3.35 m/s² -
Use the equation Vf^2 = Vo^2 + 2aY to find the distance (Y):
0^2 = 30^2 + 2 (-3.35) Y
0 = 900 – 6.7 * Y
6.7 * Y = 900
Y = 900 / 6.7
Y = 134.33 m
Verification and Sanity Check
The calculated distance is 134.33 meters. This answer is close to the “correct” answer of 132m. This is a far more reasonable distance than the original attempt’s answer of 4.475 m, which was discarded due to the unreasonably high deceleration rate.
Conclusion
When a car traveling at 30m/s runs out of gas while going uphill on a 20-degree slope, it will coast approximately 134.33 meters before starting to roll back down. This calculation highlights the importance of using the correct initial conditions, applying the proper physics principles, and performing sanity checks to ensure the solution is reasonable. Failing to appropriately utilize the angle of the slope and its effect on acceleration will lead to drastically incorrect results.
