In the javelin throw at a track-and-field event, the javelin is launched at a speed of 29 m/s at an angle of 36 degrees above the horizontal. As the javelin travels upward, it’s velocity point above the horizontal at an angle that decreases as time passes. How much time is required for the angle to be reduced from 36 degrees at launch to 18 degrees?
y = 29 sin36° t - 4.9t^2 = 17.045t - 4.9t^2
x = 29 cos36° t = 23.461t
at any point in flight, tanθ = dy/dx = (dy/dt) / (dx/dt)
So, you just need to find when
(17.045 - 9.8t)/23.461 = tan 18°
To find the time required for the angle to decrease from 36 degrees to 18 degrees, we need to use basic trigonometry and consider the projectile motion of the javelin.
Let's break down the initial velocity of the javelin into its horizontal and vertical components:
Vertical component of velocity:
Vy = V * sin(θ)
= 29 m/s * sin(36°)
= 16.88 m/s
Horizontal component of velocity:
Vx = V * cos(θ)
= 29 m/s * cos(36°)
= 23.47 m/s
The angle at which the velocity points above the horizontal decreases with time until it reaches 18 degrees.
Given that the initial angle is 36 degrees and we need to find the time required for it to reach 18 degrees, we can set up the following equation:
tan(18°) = Vy / Vx
Substituting the values we calculated earlier:
tan(18°) = 16.88 m/s / 23.47 m/s
To solve for time, we rearrange the equation:
tan(18°) = Vy / Vx
tan(18°) = (gt) / Vx [Since Vy = gt]
t = (tan(18°) * Vx) / g [Solving for t]
where g is the acceleration due to gravity, approximately 9.8 m/s².
Now, we substitute the values into the equation and calculate the time:
t = (tan(18°) * 23.47 m/s) / 9.8 m/s²
t ≈ 0.387 seconds
Therefore, it takes approximately 0.387 seconds for the angle to decrease from 36 degrees to 18 degrees.
To solve this problem, we need to understand the basic principles of projectile motion and how the angle changes with time.
Step 1: Break down the initial velocity into horizontal and vertical components.
The given initial velocity of the javelin is 29 m/s at an angle of 36 degrees above the horizontal. We can calculate the initial horizontal and vertical components of the velocity using trigonometry:
Horizontal component: Vx = V * cos(θ)
Vertical component: Vy = V * sin(θ)
Where:
V = initial velocity = 29 m/s
θ = initial angle = 36 degrees
Vx = 29 * cos(36) ≈ 23.29 m/s (horizontal component)
Vy = 29 * sin(36) ≈ 16.68 m/s (vertical component)
Step 2: Determine the time required for the angle to be reduced from 36 degrees to 18 degrees.
As the javelin travels upward, the angle above the horizontal decreases. The time taken for this angle change can be calculated using trigonometry and the vertical component of the velocity.
Let's assume the time required is 't'. At any given time, t, the vertical component of the velocity can be calculated using the initial vertical component and the acceleration due to gravity (g = 9.8 m/s^2):
Vy(t) = Vy - g * t
where:
Vy = initial vertical component = 16.68 m/s
g = acceleration due to gravity = 9.8 m/s^2
We can find the time required to reduce the angle to 18 degrees by solving for 't' in the equation:
Tan(φ) = Vy(t) / Vx
Where:
φ = angle at time 't' = 18 degrees
Rearranging the equation, we get:
Vy(t) = Vx * Tan(φ)
Substituting the values, we get:
Vy - g * t = Vx * Tan(φ)
Solving for 't', we have:
t = (Vy - Vx * Tan(φ)) / g
Substituting the given values:
t = (16.68 - 23.29 * Tan(18)) / 9.8 ≈ 1.47 seconds (rounded to two decimal places)
Therefore, it would take approximately 1.47 seconds for the angle to decrease from 36 degrees to 18 degrees.