A shot-putter launches a 7.5 kg shot with a speed of 12 m/s at an angle of 37 degrees above horizontal. If, upon release, the shot was 2.0 m above the ground, determine the horizontal range.
To find the horizontal range, we need to calculate the horizontal component of the initial velocity and the time it takes for the shot to reach the ground.
1. Calculate the horizontal component of the initial velocity:
The horizontal component of the initial velocity (Vx) can be calculated using the equation:
Vx = V * cos(θ)
where V is the initial speed and θ is the angle above the horizontal.
V = 12 m/s (given)
θ = 37 degrees (given)
Vx = 12 m/s * cos(37 degrees)
Vx ≈ 9.62 m/s
2. Calculate the time it takes for the shot to reach the ground:
The time (t) can be found using the equation:
t = (2 * V * sin(θ)) / g
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
V = 12 m/s (given)
θ = 37 degrees (given)
g = 9.8 m/s^2
t = (2 * 12 m/s * sin(37 degrees)) / 9.8 m/s^2
t ≈ 1.88 s
3. Calculate the horizontal range (R):
The horizontal range can be calculated using the equation:
R = Vx * t
Vx ≈ 9.62 m/s (calculated in step 1)
t ≈ 1.88 s (calculated in step 2)
R = 9.62 m/s * 1.88 s
R ≈ 18.1 m
Therefore, the horizontal range of the shot is approximately 18.1 meters.
To determine the horizontal range of the shot, we need to find the distance it travels horizontally before hitting the ground.
First, let's analyze the vertical motion of the shot. We need to find the time it takes for the shot to hit the ground. We can use the equation of motion:
𝑦 = 𝑣₀𝑦𝑡 + (1/2)𝑎𝑡²
Here, 𝑦 is the vertical displacement (2.0 m), 𝑣₀𝑦 is the initial vertical velocity (12 m/s * sin(37°)), 𝑎 is the acceleration due to gravity (-9.8 m/s²), and 𝑡 is the time we want to find. We can rearrange this equation to solve for 𝑡:
2.0 = (12 sin(37°))𝑡 + (1/2)(-9.8)𝑡²
Now, let's solve this equation to find the time it takes for the shot to hit the ground.
Step 1: Calculate 𝑣₀𝑦 = 12 m/s * sin(37°)
Step 2: Rewrite the equation: 2.0 = (12 sin(37°))𝑡 - 4.9𝑡²
Step 3: Rearrange the equation to standard form: 4.9𝑡² - (12 sin(37°))𝑡 + 2.0 = 0
Step 4: Solve this quadratic equation using the quadratic formula:
𝑡 = [-(−(12 sin(37°))) ± √((-(12 sin(37°)))² − 4(4.9)(2.0))] / (2)(4.9)
Simplifying further, we have:
𝑡 = [12 sin(37°) ± √((12 sin(37°))² − 4(4.9)(2.0))] / (2)(4.9)
Let's calculate the values inside the square root:
(12 sin(37°))² − 4(4.9)(2.0)
Now calculate the square root:
√((12 sin(37°))² − 4(4.9)(2.0))
After finding the values, substitute back into the equation to find 𝑡.
Once we have the time 𝑡, we can calculate the horizontal range using the equation:
𝑥 = 𝑣₀𝑥 * 𝑡
where 𝑥 is the horizontal range, and 𝑣₀𝑥 is the initial horizontal velocity (12 m/s * cos(37°)).
Let's now solve for 𝑥.