A soccer player kicks the ball toward a goal that is 23.0 m in front of him. The ball leaves his foot at a speed of 17.6 m/s and an angle of 29.0 ° above the ground. Find the speed of the ball when the goalie catches it in front of the net.
Vo = 17.6m/s[29o]
Xo = 17.6*Cos29 = 15.4 m/s
Yo = 17.6*sin29 = 8.53 m/s
Y = Yo + g*Tr = 0
Tr = -Yo/g = -8.53/-9.8 = 0.870 s. = Rise time.
Range = Dx = Vo^2*sin(2A)/g =
17.6^2*sin(58)/9.8 = 26.8 m.
h = Yo*Tr + 0.5g*Tr^2 =
8.53*0.870 - 4.9*0.870^2 = 3.71 m. = Max. ht.
Xo*(Tr+Tf) = 23 m
15.4 *(0.87+Tf) = 23
13.4 + 15.4Tf = 23
15.4Tf = 23-13.4 = 9.6
Tf = 0.624 s.
h = ho - 0.5g*Tf1^2 =
3.71 - 4.9*0.624^2 = 1.80 m. = ht. of the ball when it reaches the net.
Y^2 = Yo^2 + 2g*h = 0 + 19.6*(3.71-1.8)=
37.4
Y = 6.12 m/s = Ver. component of velocity when the ball reaches the net.
V = Xo + Yi = 15.4 + 6.12i
V^2 = 15.4^2 + 6.12^2 = 274.61
V = 16.6 m/s = Total velocity when caught by the goalie.
Note: Since the range is greater than the distance from the kicker to the goal, the ball(if not touched) will land
26.8 meters from the kicker which is 3.8
behind the net.
To find the speed of the ball when the goalie catches it, we can break down the motion of the ball into horizontal and vertical components.
Let's first find the time it takes for the ball to reach the goalie. We can use the horizontal component of the ball's motion for this.
The horizontal component of the ball's initial velocity is given by:
Vx = V * cos(theta)
where Vx is the horizontal component of the initial velocity, V is the initial velocity of the ball (17.6 m/s), and theta is the angle of projection (29.0 °).
Now, let's calculate Vx using these values:
Vx = 17.6 m/s * cos(29.0 °)
Next, we can use the horizontal component of the velocity and the distance to find the time it takes for the ball to reach the goalie.
Distance (d) = velocity * time
time = distance / velocity
Substituting the values into the equation, we have:
time = 23.0 m / Vx
Now, let's calculate the time:
time = 23.0 m / (17.6 m/s * cos(29.0 °))
Next, let's find the vertical component of the ball's velocity using the initial velocity and the angle of projection:
Vy = V * sin(theta)
where Vy is the vertical component of the initial velocity.
Now, let's calculate Vy using the given values:
Vy = 17.6 m/s * sin(29.0 °)
Now that we have the time and the vertical component of the velocity, we can find the speed of the ball when the goalie catches it. We can use the equation:
V_final = sqrt((Vx^2) + (Vy^2))
where V_final is the final speed of the ball.
Now, let's calculate V_final:
V_final = sqrt((Vx^2) + (Vy^2))
Finally, calculate V_final to find the speed of the ball when the goalie catches it.