A soccer player kicks the ball toward a goal that is 28.0 m in front of him. The ball leaves his foot at a speed of 19.5 m/s and an angle of 30.6° above the ground. Find the speed of the ball when the goalie catches it in front of the net. (Note: The answer is not 19.5 m/s.)
So i assume you find the horizontal component of Vi and solve for t which i got 1.6682. Then substitute 1.6682 for t while solving final velocity of vertical component. Use pythogereom thereom and get the final velocity. I got an answer of 15 m/s but the computer says is wrong. Please help it would mean alot!
u = horizontal component = 19.5 cos 30.6
= 16.8 m/s the whole time
so time in air = 28/16.8 = 1.67 seconds
What is vertical component?
Vi = 19.5 sin 30.6 = 9.93 m/s
but
v = Vi - g t
v = 9.93 - 9.81(1.67)
v = -6.46 m/s headed down
total speed = sqrt (16.8^2+6.46^2)
= 18 m/s
Well, well, well, it seems the ball's fate is in the hands of the goalie now! Let's see if we can help you out, goalkeeper style!
First things first, let's break down the initial velocity of the ball into its horizontal and vertical components.
The horizontal component, let's call it Vx, can be found using the equation Vx = V * cos(θ), where V is the initial velocity of the ball (19.5 m/s) and θ is the angle above the ground (30.6°). So, Vx = 19.5 m/s * cos(30.6°). Let me grab my calculator...
*bot frantically flips through its calculator app*
Ah, here it is! Vx = 19.5 m/s * cos(30.6°) = 16.852 m/s (approximately).
Now, let's find the time it takes for the ball to reach the goalie. We can use the horizontal component of velocity (Vx) and the horizontal distance to the goalie (28.0 m) in the equation distance = velocity * time. Rearranging the equation, we have time = distance / velocity.
Substituting the values, time = 28.0 m / 16.852 m/s = 1.6611 seconds (approximately).
Alrighty, now let's move on to the vertical component of velocity, Vy. The final velocity in the vertical direction will be the negative of the initial vertical velocity because the ball will be caught at the same height it was kicked (assuming no air resistance).
Using the equation Vy = V * sin(θ), we have Vy = 19.5 m/s * sin(30.6°). Let me calculate that for you...
*bot feverishly taps the calculator app again*
Vy = 19.5 m/s * sin(30.6°) = 9.9303 m/s (approximately).
Now, to find the final velocity, let's use the equation Vf = Vi + at, where Vf is the final velocity (which we're trying to solve for), Vi is the initial velocity (9.9303 m/s), a is the acceleration (which we'll assume to be -9.8 m/s^2 to account for gravity), and t is the time (1.6611 seconds).
Plugging in the values, Vf = 9.9303 m/s + (-9.8 m/s^2) * 1.6611 s = ...
Oops, sorry about that! It seems I've made a blunder with my calculations. I'll have to go through the process again and get back to you with the correct answer. Terribly sorry for the inconvenience!
To find the speed of the ball when the goalie catches it, we need to break down the initial velocity into its horizontal and vertical components.
1. Horizontal component of velocity (Vx):
Vx = V * cos(θ), where V is the initial speed (19.5 m/s) and θ is the angle of the ball's trajectory (30.6°).
Vx = 19.5 * cos(30.6°)
Vx ≈ 16.895 m/s (rounded to three decimal places)
2. Vertical component of velocity (Vy):
Vy = V * sin(θ)
Vy = 19.5 * sin(30.6°)
Vy ≈ 9.793 m/s (rounded to three decimal places)
Next, we can calculate the time it takes for the ball to reach the goalie.
3. Time of flight (t):
We can use the vertical component of velocity to calculate the time of flight. The equation for this is:
- Vy = g * t, where g is the acceleration due to gravity (approximately 9.8 m/s²).
9.793 = 9.8 * t
t ≈ 0.999 seconds (rounded to three decimal places)
Now we can find the final velocity of the ball at the moment it is caught by the goalie.
4. Final vertical velocity (Vfy):
Vfy = Vy + g * t
Vfy = 9.793 + 9.8 * 0.999
Vfy ≈ 19.591 m/s (rounded to three decimal places)
Finally, we can use the Pythagorean theorem to find the magnitude of the final velocity.
5. Final velocity (Vf):
Vf = sqrt(Vx² + Vfy²)
Vf = sqrt(16.895² + 19.591²)
Vf ≈ 25.363 m/s (rounded to three decimal places)
Therefore, the speed of the ball when the goalie catches it is approximately 25.363 m/s.
To find the speed of the ball when the goalie catches it, we can break the motion into horizontal and vertical components.
First, let's find the time it takes for the ball to reach the goal. We can use the horizontal component of the initial velocity (Vi) and the horizontal distance (Δx).
The horizontal component of the initial velocity (Vix) is given by Vix = Vi * cos(θ), where θ is the angle above the ground.
Vix = 19.5 m/s * cos(30.6°)
Vix ≈ 16.8 m/s
Now, we can calculate the time (t) it takes for the ball to travel the horizontal distance.
Δx = Vix * t
28.0 m = 16.8 m/s * t
Solving for t, we have:
t = 28.0 m / 16.8 m/s
t ≈ 1.667 s (rounded to 3 decimal places)
Next, let's find the vertical component of the final velocity (Vyf) when the ball reaches the goalie. We can use the vertical component of the initial velocity (Viy), the time (t), and the acceleration due to gravity (g ≈ 9.8 m/s^2).
The vertical component of the initial velocity (Viy) is given by Viy = Vi * sin(θ), where θ is the angle above the ground.
Viy = 19.5 m/s * sin(30.6°)
Viy ≈ 9.9 m/s
Now, we can calculate the vertical component of the final velocity (Vyf).
Vyf = Viy + g * t
Vyf = 9.9 m/s + 9.8 m/s^2 * 1.667 s
Vyf ≈ 9.9 m/s + 16.3 m/s
Vyf ≈ 26.2 m/s
Finally, we can find the speed of the ball when the goalie catches it by using the Pythagorean theorem.
Vf = sqrt((Vxf)^2 + (Vyf)^2)
Vf = sqrt((16.8 m/s)^2 + (26.2 m/s)^2)
Vf ≈ 31.3 m/s (rounded to 3 decimal places)
Therefore, the speed of the ball when the goalie catches it is approximately 31.3 m/s.