When faced with a physics problem, especially one involving motion and forces, it’s crucial to break down the scenario and apply the correct equations. Let’s analyze the situation where a car traveling at 30 m/s runs out of gas while going uphill on a 20-degree slope. The goal is to determine how far up the hill the car will coast before rolling back down.
The initial conditions are:
- Initial velocity (V₀) = 30 m/s
- Final velocity (Vf) = 0 m/s (when the car stops momentarily)
- Angle of the slope (θ) = 20 degrees
- Acceleration due to gravity (g) = 9.8 m/s²
The primary challenge here is to correctly apply the equations of motion, taking into account the effect of the slope.
Initially, the attempt focused on calculating time using: 0=30sin20 + (-9.8sin20)t. This approach incorrectly reduces the initial speed. The speed up the slope is 30m/s, not 30sin20.
Calculating the Acceleration
The component of gravitational acceleration acting along the slope is g * sin(θ). Since the car is moving uphill, the acceleration is negative (deceleration):
a = -g sin(θ) = -9.8 m/s² sin(20°) ≈ -3.35 m/s²
Determining the Distance
To find the distance the car coasts, we can use the following kinematic equation:
Vf² = Vi² + 2 a d
Where:
- Vf = Final velocity (0 m/s)
- Vi = Initial velocity (30 m/s)
- a = Acceleration (-3.35 m/s²)
- d = Distance traveled
Plugging in the values:
0² = 30² + 2 (-3.35) d
0 = 900 – 6.7 * d
- 7 * d = 900
d = 900 / 6.7 ≈ 134.3 meters
Therefore, the car will coast approximately 134.3 meters up the hill before it starts to roll back down. This calculation utilizes the correct initial speed and deceleration, providing a more accurate result. The initial attempts resulted in incorrect answers due to misapplication of trigonometric functions and misunderstanding of the problem setup.
Key Takeaways
- Sanity Checks: Always perform sanity checks on your answers. If the initial calculation suggests the car stops in 2.8 seconds, a brief thought experiment would reveal that is impossibly short, given the initial speed.
- Understanding the Physics: A car moving along a flat surface (0-degree angle) has no component of gravity slowing it down based on the sine of the angle. Ensure the equation reflects this.
- Correct Application of Equations: Using the correct kinematic equation and plugging in the values accurately are crucial for solving physics problems.
- Unit Consistency: Make sure all units are consistent throughout the calculation. Using meters for distance, meters per second for velocity, and meters per second squared for acceleration ensures a correct result in meters.
- Attention to Detail: Avoiding common mistakes, such as incorrectly applying trigonometric functions or misinterpreting initial conditions, leads to a more accurate solution.
