A softball player hits a ball at 29.5 m/s at a $33.0^{\circ}$ angle to the horizon. An outfielder catches the ball at the same height. How far was the outfielder from the hitter?
$\Delta x=[?] m$
Answer (1)
Calculate the time of flight using the formula t = g 2 v 0 s i n ( θ ) , where v 0 = 29.5 m/s, θ = 33. 0 ∘ , and g = 9.8 m/s 2 .
Calculate the horizontal component of the initial velocity: v 0 x = v 0 cos ( θ ) .
Calculate the horizontal distance using the formula Δ x = v 0 x t .
The horizontal distance between the outfielder and the hitter is $\boxed{81.12} m.
Explanation
Problem Analysis We are given the initial velocity v 0 = 29.5 m/s and the launch angle θ = 33. 0 ∘ . The ball is caught at the same height as it was hit. We need to find the horizontal distance Δ x between the hitter and the outfielder.
Calculating Time of Flight First, we need to find the time of flight t . Since the ball is caught at the same height, the vertical displacement Δ y = 0 . We can use the equation of motion for vertical displacement: Δ y = v 0 y t − 2 1 g t 2 where v 0 y = v 0 sin ( θ ) and g = 9.8 m/s 2 . Since Δ y = 0 , we have: 0 = v 0 sin ( θ ) t − 2 1 g t 2 We can factor out t : 0 = t ( v 0 sin ( θ ) − 2 1 g t ) This gives us two solutions for t : t = 0 (which is the initial time) and t = g 2 v 0 s i n ( θ ) . We are interested in the second solution, which is the time of flight.
Time of Flight Calculation Now, we can calculate the time of flight t : t = g 2 v 0 sin ( θ ) = 9.8 2 × 29.5 × sin ( 33. 0 ∘ ) Converting the angle to radians, we have θ = 33. 0 ∘ = 33.0 × 180 π ≈ 0.576 radians. Then, sin ( 33. 0 ∘ ) ≈ 0.5446 . t = 9.8 2 × 29.5 × 0.5446 ≈ 9.8 32.1434 ≈ 3.28 seco n d s
Calculating Horizontal Distance Next, we calculate the horizontal distance Δ x using the horizontal component of the initial velocity and the time of flight: Δ x = v 0 x t where v 0 x = v 0 cos ( θ ) . So, v 0 x = 29.5 × cos ( 33. 0 ∘ ) Since cos ( 33. 0 ∘ ) ≈ 0.8387 , we have v 0 x ≈ 29.5 × 0.8387 ≈ 24.74 m / s Then, Δ x = 24.74 × 3.28 ≈ 81.12 m
Final Answer Therefore, the horizontal distance between the outfielder and the hitter is approximately 81.12 meters.
Examples
Understanding projectile motion is crucial in sports like baseball or softball, where players need to accurately throw or hit the ball to a specific location. For example, knowing the initial velocity and launch angle, one can determine how far the ball will travel in the air, which helps players make strategic decisions during the game. This involves calculating the range of the projectile, which is the horizontal distance it covers before landing. By adjusting the launch angle, players can optimize the distance the ball travels, enabling them to make accurate throws or hits.
