In a football game a kicker attempts a field goal. The ball remains in contact with the kicker's foot for 0.0442 s, during which time it experiences an acceleration of 231 m/s2. The ball is launched at an angle of 55.0° above the ground. Determine the (a) horizontal and (b) vertical components of the launch velocity.
V = Vo + a*t = 0 + 231*0.0442 = 10.21 m/s.
a. V*Cos A = 10.21*Cos55 = 5.86 m/s.
b. 10.21*sin55 = 8.36 m/s.
To determine the horizontal and vertical components of the launch velocity, we can use the following kinematic equations:
For horizontal motion:
Horizontal distance (x) = Horizontal velocity (Vx) * Time (t)
For vertical motion:
Vertical distance (y) = Initial vertical velocity (Vy) * Time (t) + 0.5 * Acceleration due to gravity (g) * Time (t)^2
To calculate the horizontal component of the launch velocity (Vx), we need to find the horizontal distance (x) traveled by the ball during the time it remains in contact with the kicker's foot.
Given:
Time (t) = 0.0442 s
From the formula for horizontal motion:
x = Vx * t
Rearranging the formula:
Vx = x / t
However, we don't have the value of x directly. But we know that the initial vertical velocity (Vy) is related to the launch velocity (V) and the launch angle (θ).
Vy = V * sin(θ)
By rearranging this formula, we can solve for V:
V = Vy / sin(θ)
Now, to find the vertical component of the launch velocity (Vy), we need to calculate the vertical distance (y) traveled by the ball during the time it remains in contact with the kicker's foot.
Given:
Acceleration (a) = 231 m/s^2
Launch angle (θ) = 55.0°
From the formula for vertical motion:
y = Vy * t + 0.5 * g * t^2
Substituting the known values:
y = Vy * t + 0.5 * (-9.8 m/s^2) * t^2 (assuming g is the acceleration due to gravity, which is approximately -9.8 m/s^2)
Now, we can substitute the equation for Vy into the equation for y:
y = (V * sin(θ)) * t - 4.9 t^2
We can then rearrange this equation to isolate V:
V = (y + 4.9 t^2) / (sin(θ) * t)
Finally, we have the value of V, which we can substitute back into the equation for Vx to determine the horizontal component.
To summarize:
1. Calculate V using V = (y + 4.9 t^2) / (sin(θ) * t)
2. Calculate Vx using Vx = x / t
By following these steps, you can find the horizontal and vertical components of the launch velocity in the given football scenario.