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 35° above the horizontal. As the javelin travels upward, its velocity points above the horizontal at an angle that decreases as time passes. How much time is required for the angle to be reduced from 35° at launch to 17°?
Solve for the time t when
Y(t)/X(t) = tan 17 = 0.3057
Y = 29 sin35 * t - (g/2)t^2
X = 29 cos35 * t
0.3057 = tan 35 - 0.20643 t
Take it from there.
1.911
To find out how much time is required for the angle to be reduced from 35° at launch to 17°, we can use the concepts of projectile motion. Let's break down the problem into steps:
Step 1: Resolve the initial velocity into its horizontal and vertical components.
The initial velocity of the javelin is given as 29 m/s at an angle of 35° above the horizontal. To find the horizontal and vertical components, we can use trigonometry.
The horizontal component (Vx) can be calculated as:
Vx = V * cos(θ)
Vx = 29 m/s * cos(35°)
The vertical component (Vy) can be calculated as:
Vy = V * sin(θ)
Vy = 29 m/s * sin(35°)
Step 2: Determine the time it takes for the angle to decrease from 35° to 17°.
As the javelin travels upward, its vertical velocity decreases due to the force of gravity. At the highest point of its trajectory, the velocity is momentarily zero, and the angle decreases from 35° to 17°.
The time it takes for the javelin to reach the highest point can be calculated using the equation:
Vy = Vy(initial) - g*t
Where Vy(initial) is the initial vertical velocity (Vy), g is the acceleration due to gravity (approximately 9.8 m/s²), and t is time.
Since the vertical velocity becomes zero at the highest point, we can rearrange the equation:
0 = Vy - g*t
Next, we can calculate the time (t1) it takes for the angle to decrease from 35° to the highest point by rearranging the equation:
t1 = Vy(initial) / g
Step 3: Determine the total time it takes for the angle to decrease from 35° to 17°.
After reaching the highest point, the javelin descends back to the ground. As it descends, the angle decreases from 35° to 17°.
To find the total time, we need to consider both the ascending and descending phases of the motion. Since the time it takes for the javelin to reach the highest point is the same as the time it takes to descend from the highest point to the ground, we can calculate this time as twice the time calculated in Step 2.
t_total = 2 * t1
Substitute the value of t1 calculated earlier to find the total time it takes for the angle to decrease from 35° to 17°.