a gazelle attempts to leap a 1.8 meter fence. assuming a 45 degree takeoff angle, what's the minimum speed for the jump?
recall that for a projectile motion, the maximum height that the object can reach is given by the formula,
h,max = [(vo)^2 * sin^2 (θ)]/(2g)
where
vo = initial velocity
θ = angle of release
g = acceleration due to gravity = 9.8 m/s^2
the unknown in the problem is vo. thus we just substitute the given in the formula:
h,max = [(vo)^2 * sin^2 (θ)]/(2g)
1.8 = [(vo)^2 * sin^2 (45)]/(2*9.8)
1.8*2*9.8 = (vo)^2 * (1/2)
34.2 = (vo)^2 * (1/2)
70.56 = (vo)^2
vo = sqrt(70.56)
vo = 8.4 m/s
hope this helps~ :)
To calculate the minimum speed required for a gazelle to leap a 1.8-meter fence at a 45-degree takeoff angle, we can use basic principles of projectile motion. Let's break down the steps:
Step 1: Convert the takeoff angle from degrees to radians.
- 45 degrees * (π/180) = 0.7854 radians
Step 2: Calculate the vertical component of velocity using the initial takeoff angle.
- Vertical component velocity (Vy) = speed * sin(angle)
Step 3: Calculate the time it takes for the gazelle to reach its highest point (time of flight).
- Time of flight = 2 * (Vy / gravity)
- Assume gravity to be -9.8 m/s^2 (taking downward as the negative direction)
Step 4: Calculate the horizontal distance travelled during the jump.
- Horizontal distance = speed * cos(angle) * time of flight
Step 5: Set the minimum horizontal distance equal to the height of the fence and solve for the minimum speed.
- Horizontal distance = 1.8 meters
By following these steps, we can calculate the minimum speed required for the gazelle to clear the fence.
To calculate the minimum speed required for the gazelle to leap over the fence, we can use the principles of projectile motion. Here's how you can do it step by step:
1. Identify the given information:
- Initial angle of takeoff: 45 degrees (θ)
- Height of the fence: 1.8 meters (h)
2. Determine the vertical and horizontal components of the initial velocity (v):
- Vertical component: v * sin(θ)
- Horizontal component: v * cos(θ)
3. Calculate the time it takes for the gazelle to reach the maximum height (t1):
- We know that the vertical component of the velocity becomes zero at the highest point of the projectile motion. Therefore, we can use the equation:
v * sin(θ) - g * t1 = 0
Where g is the acceleration due to gravity (approximately 9.8 m/s^2).
- Rearranging the equation, we get:
t1 = v * sin(θ) / g
4. Calculate the time it takes for the gazelle to fall from the maximum height to the height of the fence (t2):
- Using the equation for vertical displacement in projectile motion, we have:
h = (1/2) * g * t2^2
- Rearranging the equation:
t2^2 = 2h / g
t2 = sqrt(2h / g)
(Note: We use the positive square root because time is always positive.)
5. The total time of flight (t_total) is the sum of t1 and t2:
t_total = t1 + t2
6. The horizontal distance covered by the gazelle during the jump can be calculated using:
- Horizontal component of velocity * t_total
Now that we have the steps, let's substitute the given values and calculate the minimum speed for the jump:
1. Calculate t1:
t1 = v * sin(45) / g = (v / sqrt(2)) / 9.8
2. Calculate t2:
t2 = sqrt(2 * 1.8 / 9.8) = sqrt(0.3673) ≈ 0.606 s
3. Calculate t_total:
t_total = t1 + t2 = (v / sqrt(2)) / 9.8 + 0.606
4. Calculate the horizontal distance:
Horizontal distance = v * cos(45) * t_total = (v / sqrt(2)) * (v / sqrt(2)) / 9.8 + 0.606
5. For the gazelle to clear the fence, the horizontal distance should be greater than or equal to the width of the fence. Assuming the width of the fence is negligible compared to the horizontal distance, we can approximate the horizontal distance as:
Horizontal distance ≈ v^2 / (9.8 * 2) + 0.606
6. Finally, solve for v:
v^2 / 19.6 + 0.606 ≥ 0.0
v^2 / 19.6 ≥ -0.606
v^2 ≥ -0.606 * 19.6
v^2 ≥ -11.8624
Since the minimum speed (v) must be a positive value, we can conclude that the minimum speed required for the gazelle to clear the 1.8-meter fence is greater than 0.