A football is thrown upward at 34 degrees angle to the horizontal. The acceleration of gravity is 9n8. To throw the ball a distance 66.7m what must be the initial speed of the ball
To find the initial speed of the ball, we can use the projectile motion equations.
The horizontal and vertical components of the initial velocity can be found using the given angle and the total initial velocity.
Given:
Angle (θ) = 34 degrees
Distance (d) = 66.7 m
Acceleration due to gravity (g) = 9.8 m/s^2
Step 1: Find the horizontal component of the initial velocity (Vx).
We know that Vx = V * cos(θ), where V is the initial velocity.
Vx = V * cos(34 degrees).
Step 2: Find the time of flight (t) for the projectile.
Since the vertical displacement is zero at the highest point, we can use the equation:
0 = Vy*t - (1/2) * g * t^2.
Step 3: Determine the vertical component of the initial velocity (Vy).
We know that Vy = V * sin(θ), where V is the initial velocity.
Vy = V * sin(34 degrees).
Step 4: Use the horizontal component (Vx) and the time of flight (t) to find the total horizontal distance (dx) covered by the projectile.
dx = Vx * t.
Step 5: Set the total horizontal distance equal to the given distance and solve for the initial velocity V.
dx = 66.7 m.
Finally, we can summarize the steps and solve for V:
Vx = V * cos(34 degrees).
Vy = V * sin(34 degrees).
0 = Vy * t - (1/2) * g * t^2.
dx = Vx * t.
dx = 66.7 m.
Now we can proceed to solve for the initial velocity V.
To find the initial speed of the ball, we can break down the given information and use the equations of motion.
First, we need to determine the time it takes for the ball to reach its peak height. At the maximum height, the vertical component of velocity becomes zero. We can use the equation:
v = u + at
where:
v = final velocity (0 m/s at maximum height)
u = initial velocity (unknown)
a = acceleration due to gravity (-9.8 m/s^2)
t = time
Since the ball is thrown upward, we can consider the acceleration due to gravity as negative. Therefore, the equation becomes:
0 = u - 9.8t
Next, we can determine the time it takes for the ball to reach the peak. The vertical velocity at the peak is given by:
v = u + at
where:
v = final velocity (0 m/s at maximum height)
u = initial velocity (unknown)
a = acceleration due to gravity (-9.8 m/s^2)
t = time
Substituting the values:
0 = u - 9.8t
Solving for t:
t = u / 9.8
Now, we can determine the total time the ball is in the air. Since the ball is thrown upward, it will take the same amount of time to reach its peak as it takes to descend back to the starting height. Therefore, the total time in the air is:
2t = 2u / 9.8
Finally, we can calculate the horizontal distance traveled by the ball using the formula:
d = ut + (1/2)at^2
where:
d = horizontal distance (66.7 m)
u = initial velocity (unknown)
t = total time in the air (2u / 9.8)
Substituting the values:
66.7 = (u * (2u / 9.8)) + (1/2)(-9.8)((2u / 9.8)^2)
Simplifying and solving the equation will give us the initial velocity (u) needed to throw the ball a distance of 66.7 m.