a gazelle attempts to leap a 2.1m fence. assuming a 45 degree takeoff angle, whats the min speed for the jump
To calculate the minimum speed needed for the gazelle to successfully leap over a 2.1m fence at a 45-degree takeoff angle, we can use kinematic equations.
The height of the fence, h, is 2.1m. We know that the initial vertical velocity, vy, is related to the initial speed, v0, and the takeoff angle, θ, by the equation vy = v0 * sin(θ).
Since the gazelle's takeoff angle is 45 degrees, we can substitute sin(45) with (√2)/2:
vy = v0 * (√2)/2
Now, let's consider the vertical motion of the gazelle. The time it takes to reach maximum height can be calculated using the equation:
t = vy / g
Where g is the acceleration due to gravity, approximately 9.8m/s².
The time it takes for the gazelle to reach maximum height is the same as the time it takes to fall back down to the ground. Thus, the total time of flight, T, is twice the time to reach maximum height.
T = 2t = 2(vy / g)
To calculate the minimum speed required for the jump, we need to find the horizontal component of the velocity, vx. The horizontal distance, x, traveled by the gazelle can be found using the equation:
x = vx * T
Since the gazelle wants to clear the 2.1m fence, the vertical distance traveled, y, is equal to the height of the fence:
y = h = vy * T - (0.5 * g * T²)
We can now substitute the values we have calculated:
2.1 = (v0 * (√2)/2) * (2(v0 * (√2)/2) / 9.8) - (0.5 * 9.8 * (2(v0 * (√2)/2) / 9.8)²)
Simplifying this equation will give us the minimum speed needed for the gazelle to jump over the fence at a 45-degree takeoff angle.
To find the minimum speed required for a gazelle to leap a 2.1m fence at a 45-degree takeoff angle, you can use the principles of projectile motion. Let's break it down step by step:
Step 1: Identify the information given:
- Fence height (h): 2.1m
- Takeoff angle (θ): 45 degrees
Step 2: Analyze the problem:
When the gazelle leaps, its motion can be divided into two separate components: a vertical component (up and down) and a horizontal component (forward motion). The vertical component determines the height the gazelle can reach, while the horizontal component determines the distance it can leap.
Step 3: Solve for the vertical component:
The vertical component of the gazelle's velocity can be calculated using the following formula:
Vy = V * sin(θ)
where:
- Vy is the vertical component of velocity
- V is the initial velocity of the gazelle
- θ is the takeoff angle (45 degrees)
To clear the fence's height, the vertical component of the gazelle's velocity must be equal to or greater than the height of the fence (2.1m). Therefore, we have:
Vy = V * sin(45°) ≥ 2.1m
Step 4: Solve for the horizontal component:
The horizontal component of the gazelle's velocity can be calculated using the same formula as before, but with the cosine function instead:
Vx = V * cos(θ)
where:
- Vx is the horizontal component of velocity
- V is the initial velocity of the gazelle
- θ is the takeoff angle (45 degrees)
Step 5: Calculate the minimum speed:
Since the vertical velocity component is the one that determines if the gazelle can clear the fence, we can focus on it. Rearranging the formula from Step 3, we get:
V ≥ 2.1m / sin(45°)
Using the value of sin(45°) ≈ 0.707, we can substitute it into the equation:
V ≥ 2.1m / 0.707
V ≥ 2.97m/s
Therefore, the minimum speed required for the gazelle to clear the 2.1m fence at a 45-degree takeoff angle is approximately 2.97 meters per second.