6. To win a javelin throwing competition, your last
throw must travel a horizontal distance of at least
100 feet. You release the javelin at a 40° angle with
an initial speed of 71 feet per second. Do you win the
competition? Justify your answer
To determine if you win the competition, we need to find the horizontal distance covered by the javelin in its last throw.
The horizontal distance covered by a projectile can be calculated using the formula:
horizontal_distance = initial_velocity * time * cos(angle)
Given:
- Initial velocity (v0) = 71 feet per second
- Launch angle (θ) = 40°
First, we need to find the time it takes for the javelin to reach the ground. The time it takes for a projectile to reach its highest point is equal to half of the total time of flight.
We can find the total time of flight using the formula:
time_of_flight = (2 * initial_velocity * sin(angle)) / acceleration_due_to_gravity
Given:
- Acceleration due to gravity (g) = 32.2 ft/s² (approximately)
Plugging in the values, we can calculate the time of flight:
time_of_flight = (2 * 71 * sin(40°)) / 32.2
≈ 3.40 seconds
Now, we can find the horizontal distance covered by the javelin:
horizontal_distance = 71 * 3.40 * cos(40°)
≈ 204.83 feet
The horizontal distance covered by the javelin in its last throw is approximately 204.83 feet. Since this distance exceeds the required 100 feet, you win the competition.
To determine whether you win the javelin throwing competition, we need to analyze the motion of the javelin and calculate the horizontal distance it travels.
The motion of the javelin can be divided into horizontal and vertical components. The horizontal component determines the distance covered, while the vertical component affects the height the javelin reaches.
Let's break down the given information:
Initial speed (v0) = 71 feet per second
Launch angle (θ) = 40°
First, we need to calculate the initial horizontal velocity (Vx) and the initial vertical velocity (Vy).
Vx = v0 * cosθ
Vy = v0 * sinθ
Vx = 71 * cos(40°)
Vx ≈ 71 * 0.766
Vx ≈ 54.386 feet per second
Vy = 71 * sin(40°)
Vy ≈ 71 * 0.643
Vy ≈ 45.713 feet per second
Now, let's calculate the time it takes for the javelin to reach the highest point of its trajectory (time to reach maximum height, t_max). At this point, the vertical velocity becomes zero.
Using the equation for vertical velocity:
Vy = Vy0 - g * t_max
0 = 45.713 - 32.2 * t_max
t_max ≈ 1.4207 seconds
Next, we can find the total flight time of the javelin, which is twice the time it takes to reach the maximum height.
Total time of flight (t_total) = 2 * t_max
t_total = 2 * 1.4207 ≈ 2.8414 seconds
Now, we can calculate the horizontal distance traveled (d) by multiplying the horizontal velocity (Vx) by the total flight time (t_total).
d = Vx * t_total
d = 54.386 * 2.8414
d ≈ 154.51 feet
According to the calculations, the horizontal distance traveled by the javelin is approximately 154.51 feet, which is greater than the required distance of 100 feet. Therefore, you win the competition as your last throw exceeds the minimum distance required.