Calculating Acceleration: Car On A Frictionless Hill

Oct 23, 2025 by SLV Team 53 views

Hey there, physics enthusiasts! Today, we're diving into a classic problem involving forces, motion, and a bit of trigonometry. Imagine a 1500 kg car parked at the top of a frictionless hill. This car is just chilling, right at rest, until, well, gravity decides to have some fun. The hill is inclined at a sweet $25.8^{\circ}$ angle. Our mission? To figure out the acceleration of this car as it gracefully slides down the hill. Sounds like a fun challenge, doesn't it?

This isn't just about crunching numbers; it's about understanding how forces interact and how they influence an object's motion. This problem beautifully illustrates Newton's Second Law of Motion, the foundation of classical mechanics. We'll break down the forces at play, apply some trigonometric magic, and, boom, we'll have our answer. The best part? No complicated equations or complex calculus needed! Just some basic physics principles and a touch of problem-solving skills will get us there. So, buckle up, and let's get rolling!

Unveiling the Forces: A Deep Dive

Alright, let's get down to the nitty-gritty and analyze the forces acting on the car. Since the hill is frictionless, we can skip the hassle of dealing with friction. So, what forces are at play? Only two main ones: the force of gravity and the normal force.

The Force of Gravity

First off, we have gravity. This force is pulling the car downwards, right towards the center of the Earth. The magnitude of this force is given by $F_g = mg$ , where $m$ is the mass of the car (1500 kg) and $g$ is the acceleration due to gravity (approximately $9.8 \ m/s^2$ ). But here's the kicker: we need to break this force down into its components because the car is on an incline. Why? Because the car is not simply falling straight down; it's sliding down the hill. We'll soon find the component of gravity that's parallel to the hill's surface is the one causing the acceleration. This is where a little bit of trigonometry comes in handy.

The Normal Force

Next, we have the normal force. This is the force exerted by the hill on the car, pushing perpendicularly to the hill's surface. It's the hill's way of preventing the car from sinking into it. In our case, since the hill is not accelerating and is assumed to be stationary, the normal force will equal the component of the gravitational force that is perpendicular to the hill. It is very important to mention that the normal force does not affect the car's acceleration down the hill. It counteracts the perpendicular component of gravity.

To summarize: The car experiences the force of gravity and the normal force. We only need the component of gravity that acts parallel to the slope to determine the car's acceleration. With a complete understanding of the forces involved, we can now move on to the next step, where we apply Newton's second law.

Setting Up the Equations: Putting the Pieces Together

Now comes the fun part: let's set up the equations and get those calculations done. Remember Newton's Second Law? It states that $F_{net} = ma$ , where $F_{net}$ is the net force acting on the object, $m$ is its mass, and $a$ is its acceleration. In our scenario, the net force is the component of gravity that acts down the hill, which we'll call $F_{gx}$ . So, the equation becomes $F_{gx} = ma$ .

Calculating the Parallel Component of Gravity

To find $F_{gx}$ , we'll use trigonometry. Since the hill is inclined at $25.8^{\circ}$ , the angle between the gravitational force ( $F_g$ ) and the normal force is also $25.8^{\circ}$ (geometry is awesome!). The parallel component of gravity, $F_{gx}$ , can be calculated using the following formula:

$F_{gx} = F_g \cdot sin(\theta) = mg \cdot sin(\theta)$

Where $\theta$ is the angle of inclination, $m$ is the mass of the car, and $g$ is the acceleration due to gravity.

Let's plug in the numbers:

$F_{gx} = (1500 \ kg) \cdot (9.8 \ m/s^2) \cdot sin(25.8^{\circ})$

$F_{gx} \approx 6489.6 \ N$

Solving for Acceleration

Now, we can use Newton's Second Law, $F_{gx} = ma$ , to solve for the acceleration, $a$ .

$a = \frac{F_{gx}}{m}$

$a = \frac{6489.6 \ N}{1500 \ kg}$

$a \approx 4.33 \ m/s^2$

So, the acceleration of the car down the hill is approximately $4.33 \ m/s^2$ . This means that the car will increase its speed by 4.33 meters per second every second as it slides down the frictionless hill. Pretty cool, huh? But always remember that these calculations assume an idealized scenario: a frictionless hill.

Conclusion: Wrapping It Up and Key Takeaways

There you have it, folks! We've successfully calculated the acceleration of the car sliding down the frictionless hill. We started with the basics, broke down the forces, applied Newton's Second Law, and used some trigonometry to get our answer. Remember, the key is to understand the forces involved, break them down into components where necessary, and apply the appropriate physics principles.

Key Points to Remember

Gravity's Role: Gravity is the driving force behind the car's acceleration. However, only the component of gravity parallel to the hill affects the car's motion.
Newton's Second Law: $F_{net} = ma$ is your best friend when dealing with forces and motion.
Trigonometry's Magic: Use sine and cosine to break down forces into their components, especially on inclined planes.
Idealized Scenario: The frictionless hill is a simplification. Real-world scenarios involve friction, which would reduce the car's acceleration.

This exercise highlights the power of physics in understanding the world around us. By breaking down complex problems into manageable steps, we can solve them with relative ease.

So, the next time you see a car sliding down a hill (or any object in motion), remember the principles we discussed today. Keep exploring, keep questioning, and most importantly, keep having fun with physics!

I hope this helped. If you have any more questions or want to tackle another physics problem, feel free to ask. Stay curious, and keep exploring the amazing world of physics!