A goalkeeper kicks the ball from his goal area at a velocity of 20 m/s and at an angle of 31° to the horizontal. Calculate: 4.1 The horizontal displacement of the stone.

Answer 1

To find the horizontal displacement of the ball, we first need to break down the initial velocity into its horizontal and vertical components.

The horizontal component of the velocity (Vx) can be found using the formula:
Vx = V * cos(θ)
where V is the magnitude of the velocity (20 m/s) and θ is the angle of 31°.

Vx = 20 m/s * cos(31°)
Vx = 20 m/s * 0.857
Vx = 17.14 m/s

Now that we have calculated the horizontal component of the velocity, we can calculate the time it takes for the ball to reach the ground. We can use the vertical component of the velocity (Vy) to do this.

The vertical component of the velocity (Vy) can be found using the formula:
Vy = V * sin(θ)
where V is the magnitude of the velocity (20 m/s) and θ is the angle of 31°.

Vy = 20 m/s * sin(31°)
Vy = 20 m/s * 0.516
Vy = 10.32 m/s

Next, we can calculate the time it takes for the ball to reach the ground using the vertical component of the velocity:
-10 m/s = 10.32 m/s - 9.81 m/s^2 * t
-10.32 m/s / 9.81 m/s^2 = t
t ≈ 1.05 s

Finally, we can calculate the horizontal displacement by multiplying the time by the horizontal component of the velocity:
Horizontal displacement = Vx * t
Horizontal displacement = 17.14 m/s * 1.05 s
Horizontal displacement ≈ 18 m

Therefore, the horizontal displacement of the ball is approximately 18 meters.