A football is thrown upward at a(n) 23◦ angleto the horizontal.
The acceleration of gravity is 9.8 m/s
To throw a(n) 52 m pass, what must be the
initial speed of the ball?
Answer in units of m/s
To find the initial speed of the ball, we can use the kinematic equation for projectile motion:
y = y0 + v0y*t - 1/2*g*t^2
Where:
y = vertical displacement (52 m in this case)
y0 = initial vertical position (0 m since we're starting at ground level)
v0y = initial vertical velocity (unknown)
g = acceleration due to gravity (-9.8 m/s^2)
t = time of flight (unknown)
First, let's find the time of flight. The total time of flight can be calculated using the formula:
t = 2*v0y/g
Plugging in the values, we have:
52 = 2*v0y*(-9.8)
52 = -19.6*v0y
v0y = -52/19.6
v0y ≈ -2.656 m/s
Since the ball is thrown upward, the vertical velocity (v0y) is negative. However, we need to take the magnitude of the velocity, so we ignore the negative sign.
Now, let's find the initial speed of the ball using the same time of flight. The horizontal displacement is given by:
x = v0x*t
Since we know the angle of projection (23°) and the initial vertical velocity (v0y), we can calculate the initial horizontal velocity (v0x) using trigonometry:
v0x = v0 * cos(θ)
Here, θ is the angle of projection and v0 is the initial speed (unknown).
To find the initial speed (v0), we need to calculate v0x using the above equation:
v0x = v0 * cos(23°)
Now, substituting this back into the horizontal displacement equation:
52 = v0 * cos(23°) * t
We can rearrange this equation to find v0:
v0 = 52 / (cos(23°) * t)
Substituting the value of v0y from earlier and rearranging:
v0 = 52 / (cos(23°) * (2 * v0y / g))
v0 = 52 * g / (2 * cos(23°) * v0y)
Finally, substituting the known values and calculating:
v0 = 52 * 9.8 / (2 * cos(23°) * 2.656)
v0 ≈ 31.97 m/s
Therefore, the initial speed of the ball must be approximately 31.97 m/s.
To find the initial speed of the ball, we can use the following steps:
1. Resolve the initial velocity of the ball into its horizontal and vertical components.
The vertical component of the initial velocity can be found using the formula:
Vyi = Vi * sin(θ)
where Vyi is the vertical component of the initial velocity, Vi is the initial velocity, and θ is the angle of the trajectory (in this case, 23 degrees).
2. Since the ball is thrown directly upward, the horizontal component of the initial velocity is zero.
3. Calculate the time it takes for the ball to reach its maximum height using the formula:
t = Vyi / g
where t is the time, Vyi is the vertical component of the initial velocity, and g is the acceleration due to gravity (9.8 m/s²).
4. Determine the maximum height reached by the ball using the formula for vertical displacement:
H = Vyi^2 / (2 * g)
where H is the maximum height, Vyi is the vertical component of the initial velocity, and g is the acceleration due to gravity.
5. Calculate the time it takes for the ball to descend from the maximum height to the ground using the formula:
t = sqrt(2 * H / g)
where t is the time, H is the maximum height, and g is the acceleration due to gravity.
6. Determine the total time of flight by doubling the time it took for the ball to reach its maximum height:
t_total = 2 * t
7. Calculate the horizontal distance traveled by the ball using the formula:
d = Vx * t_total
where d is the horizontal distance, Vx is the horizontal component of the initial velocity, and t_total is the total time of flight.
8. Rearrange the equation to solve for the initial velocity (Vi), knowing that the horizontal component of the initial velocity (Vx) is zero:
d = Vi * cos(θ) * t_total
Simplifying the equation gives us:
Vi = d / (cos(θ) * t_total)
Now we can apply these steps and find the initial speed of the ball:
Given:
θ = 23 degrees
d = 52 m
g = 9.8 m/s²
1. Find the vertical component of the initial velocity:
Vyi = Vi * sin(θ)
Vyi = Vi * sin(23◦)
2. The horizontal component of the initial velocity is zero.
3. Calculate the time it takes for the ball to reach its maximum height:
t = Vyi / g
4. Determine the maximum height reached by the ball:
H = Vyi^2 / (2 * g)
5. Calculate the time it takes for the ball to descend from the maximum height to the ground:
t = sqrt(2 * H / g)
6. Determine the total time of flight:
t_total = 2 * t
7. Calculate the horizontal distance traveled:
d = Vx * t_total (Note: Vx is zero in this case)
8. Solve for the initial velocity (Vi):
Vi = d / (cos(θ) * t_total)
By following these steps and substituting the given values, we can find the initial speed of the ball.