4. A ball is thrown so that it just clears a 3m fence 18m away. If it left the hand 1.5m above the ground and at an angle of 60° with the horizontal, what was the initial velocity of the ball?
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To find the initial velocity of the ball, we can use the equations of projectile motion.
Let's break down the given information:
- The ball is thrown at an angle of 60° with the horizontal.
- It clears a 3m fence located 18m away.
- The ball leaves the hand 1.5m above the ground.
We can start by analyzing the vertical component of the ball's motion. At the highest point of its trajectory, the ball's vertical velocity component will be zero. We can use this information to find the time it takes for the ball to reach its maximum height:
1. Use the formula for vertical displacement:
Δy = v₀y * t + 0.5 * g * t^2
Since Δy = 1.5m and v₀y = v₀ * sinθ (where v₀ is the initial velocity and θ is the launch angle), we can rewrite the equation as:
1.5 = (v₀ * sin60°) * t + 0.5 * (-9.8) * t^2
2. Solve the equation for t:
Set the equation equal to zero:
0.5 * (-9.8) * t^2 + (v₀ * sin60°) * t - 1.5 = 0
This quadratic equation can be solved using the quadratic formula.
Next, we can analyze the horizontal component of the ball's motion. The horizontal distance traveled by the ball is 18m. We can use this information to find the time it takes for the ball to travel this distance:
3. Use the formula for horizontal displacement:
Δx = v₀x * t
Since Δx = 18m and v₀x = v₀ * cosθ, we can rewrite the equation as:
18 = (v₀ * cos60°) * t
4. Rearrange the equation to solve for t:
t = 18 / (v₀ * cos60°)
Now, we have expressions for t (from step 2) and t (from step 4). We can equate them to solve for the initial velocity, v₀:
5. Equate the expressions for t from step 2 and t from step 4:
0.5 * (-9.8) * t^2 + (v₀ * sin60°) * t - 1.5 = 18 / (v₀ * cos60°)
6. Rearrange the equation to solve for v₀:
Simplify the equation and solve for v₀.
By following these steps, you can find the initial velocity of the ball.
Range = Vo^2*sin(2A)/g = 18
A = 60o
g = 9.8 m/s^2
Vo = Initial velocity in m/s.
Solve for Vo.