A projectile is launched off the roof of a 180-meter-tall building close to its edge, where the initial velocity's horizontal component is [tex]$20 m / s$[/tex] and its vertical component is [tex]$25 m / s$[/tex]. The projectile lands on the ground below. How far from the building's side (in meters) does the projectile land? Round your answer to the nearest hundredth (0.01).
Answer (1)
Determine the time the projectile spends in the air using the vertical motion equation: Δ y = v 0 y t + 2 1 a t 2 .
Solve the quadratic equation 4.9 t 2 − 25 t − 180 = 0 using the quadratic formula to find the time t ≈ 9.1269 seconds.
Calculate the horizontal distance using the formula: distance = horizontal velocity * time, which gives d i s t an ce = 20 × 9.1269 ≈ 182.538 meters.
Round the final distance to the nearest hundredth: 182.54
Explanation
Problem Setup We are given that a projectile is launched from a 180-meter-tall building with an initial horizontal velocity of 20 m/s and an initial vertical velocity of 25 m/s. Our goal is to find the horizontal distance the projectile travels before landing on the ground.
Vertical Motion Analysis First, we need to determine the time the projectile spends in the air. We can use the vertical motion to find this time. The vertical displacement is -180 meters (since the projectile lands below the launch point). We use the equation of motion: Δ y = v 0 y t + 2 1 a t 2 where Δ y = − 180 , v 0 y = 25 , and a = − 9.8 .
Setting up the Quadratic Equation Substituting the values, we get: − 180 = 25 t − 4.9 t 2 Rearranging the equation, we have: 4.9 t 2 − 25 t − 180 = 0
Applying the Quadratic Formula Now, we use the quadratic formula to solve for t: t = 2 a − b ± b 2 − 4 a c where a = 4.9 , b = − 25 , and c = − 180 .
Calculating the Time Plugging in the values, we get: t = 2 ( 4.9 ) 25 ± ( − 25 ) 2 − 4 ( 4.9 ) ( − 180 ) t = 9.8 25 ± 625 + 3528 t = 9.8 25 ± 4153 t = 9.8 25 ± 64.44377
Choosing the Correct Time We have two possible values for t: t 1 = 9.8 25 + 64.44377 = 9.8 89.44377 ≈ 9.1269 t 2 = 9.8 25 − 64.44377 = 9.8 − 39.44377 ≈ − 4.0249 Since time cannot be negative, we choose the positive value: t ≈ 9.1269 seconds.
Calculating the Horizontal Distance Now, we calculate the horizontal distance using the formula: distance = horizontal velocity * time. The horizontal velocity is 20 m/s, so: d i s t an ce = 20 × 9.1269 = 182.538
Final Answer Rounding to the nearest hundredth, the horizontal distance is approximately 182.54 meters.
Examples
Understanding projectile motion is crucial in fields like sports and military science. For example, when a basketball player shoots a ball, they intuitively calculate the angle and initial velocity needed to make the shot. Similarly, in military applications, accurately predicting the landing point of a projectile is essential for targeting. The principles used here help in designing artillery and other ballistic systems.
