A ditch 2.5 wide crosses a trail bike path (see the figure(Figure 1) ). An upward incline of 15 has been built on the approach so that the top of the incline is level with the top of the ditch.
What is the minimum speed a trail bike must be moving to clear the ditch? (Add 1.7 to the range for the back of the bike to clear the ditch safely.)
Well, it seems we have a biking stunt in the making! To clear the ditch safely, we need to calculate the minimum speed the trail bike must be moving.
Now, the width of the ditch is 2.5 meters. And because we want to add a safe margin of 1.7 meters for the back of the bike, that brings our total distance to clear to 4.2 meters.
To simplify things, let's forget about the incline for a moment and focus on launching the bike horizontally. In that case, we can use the formula: distance = speed × time.
Since we're concerned with the minimum speed, we'll assume the bike is launched at a 45-degree angle. Why? Because a 90-degree angle would be too vertical, and a 0-degree angle would be too horizontal. So, let's compromise. That's how relationships work!
Now, when a projectile is launched at a 45-degree angle, physics tells us that the horizontal distance traveled is equal to the initial velocity (speed) multiplied by the time of flight.
To find the time of flight, we can use the formula: time = 2 × initial velocity × sin(angle) / gravity.
The acceleration due to gravity on Earth is approximately 9.8 meters per second squared. So, with a bit of math and a sprinkle of trigonometry, we'll find our desired minimum speed.
But remember, safety first! Stay within your limits and consider wearing a clown helmet while attempting this stunt. Safety and entertainment combined - what more could you ask for?
To find the minimum speed a trail bike must be moving to clear the ditch, we need to consider the physics involved in this scenario.
First, let's analyze the situation. We have a ditch that is 2.5 meters wide and an upward incline of 15 degrees that has been built on the approach. The top of the incline is level with the top of the ditch. We also know that an additional 1.7 meters of range is required for the back of the bike to safely clear the ditch.
To clear the ditch, the bike needs to have enough horizontal velocity to cover the width of the ditch during the time it takes for the bike to reach the other side. The time it takes for an object to travel a certain distance is determined by its initial velocity and the acceleration acting on it.
Let's break down the problem into smaller steps:
Step 1: Calculate the time it takes for the bike to reach the other side of the ditch.
To do this, we need to find the acceleration of the bike. The only horizontal force acting on the bike is the component of its weight parallel to the incline. This force can be calculated using the formula: F = m * g * sin(θ), where m is the mass of the bike, g is the acceleration due to gravity, and θ is the angle of the incline.
Step 2: Calculate the required initial velocity of the bike.
To cover the width of the ditch, the bike needs to travel a distance of 2.5 meters. We can use the formula: d = v * t, where d is the distance, v is the initial velocity, and t is the time.
Step 3: Add the required additional range.
We need to add the extra 1.7 meters of range to the distance traveled by the bike in order to ensure that the back of the bike clears the ditch safely. Therefore, we need to consider a total distance traveled of 2.5 meters + 1.7 meters = 4.2 meters.
Step 4: Calculate the minimum speed required.
Using the formula: v = d / t, where v is the initial velocity, d is the distance, and t is the time, we can calculate the minimum speed required.
To summarize, to find the minimum speed a trail bike must be moving to clear the ditch:
1. Calculate the time it takes for the bike to reach the other side of the ditch using the acceleration due to the incline.
2. Calculate the required initial velocity of the bike.
3. Add the required additional range to the distance traveled by the bike.
4. Calculate the minimum speed required using the distance and time calculated in steps 2 and 3.
Note: To provide an accurate answer, the mass of the bike and the acceleration due to gravity are needed, as they affect the calculations.