A soccer ball is kicked with an initial velocity of v₀ = 25.0 m/s at an angle of 35.0° with respect to the horizontal. How far does the soccer ball travel horizontally, in meters, before hitting the ground?
Round your answer to 3 significant figures.
Answer (2)
The soccer ball travels approximately 60.0 meters horizontally before hitting the ground. This calculation involves determining the horizontal and vertical components of the initial velocity, calculating the time of flight, and then finding the horizontal distance. Rounding this value gives us the final answer.
;
To solve this problem, we need to calculate how far a soccer ball travels horizontally before it hits the ground. This is a problem involving projectile motion, where the ball is kicked at an initial velocity and angle.
First, let's break down the initial velocity of the soccer ball into its horizontal and vertical components:
Horizontal component of velocity (vₓ):
v x = v 0 ⋅ cos ( θ )
v x = 25.0 m/s ⋅ cos ( 35.0° ) ≈ 20.48 m/s
Vertical component of velocity (vᵧ):
v γ = v 0 ⋅ sin ( θ )
v γ = 25.0 m/s ⋅ sin ( 35.0° ) ≈ 14.34 m/s
Next, we need to determine how long the soccer ball is in the air. This is the time it takes for the vertical velocity to bring the ball back to the ground level (ignoring air resistance).
Time of flight (t):
The vertical motion can be described by the kinematic equation:
t = g 2 ⋅ v γ
Where:
g is the acceleration due to gravity (approximately 9.81 m/s²)
t = 9.81 m/s 2 2 ⋅ 14.34 m/s ≈ 2.92 s
Finally, we can calculate the horizontal distance traveled (range) using the horizontal velocity and the time of flight:
Horizontal distance (R):
R = v x ⋅ t
R = 20.48 m/s ⋅ 2.92 s ≈ 59.8 m
Thus, the soccer ball travels approximately 59.8 meters horizontally before hitting the ground. Note that the final result is rounded to three significant figures.
