A spider crawling across a table leaps onto a magazine blocking its path. The initial velocity of the spider is 0.850 m/s at an angle of 34.3° above the table, and it lands on the magazine 0.0780 s after leaving the table. Ignore air resistance. How thick is the magazine? Express your answer in millimeters.

Vi up = .85 sin 34.3 = .479 m/s

h = 0 + Vi t - 4.9 t^2

h = .479(.078) - 4.9(.078)^2

in meters, multiply by 1000 to get mm

To determine the thickness of the magazine, we can break down the motion of the spider into horizontal and vertical components.

First, let's find the horizontal component of the spider's initial velocity (Vx). We can use the cosine function to calculate this:

Vx = V * cos(θ)
Vx = 0.850 m/s * cos(34.3°)
Vx = 0.850 m/s * 0.822
Vx = 0.6982 m/s

Next, let's find the vertical component of the spider's initial velocity (Vy). We can use the sine function to calculate this:

Vy = V * sin(θ)
Vy = 0.850 m/s * sin(34.3°)
Vy = 0.850 m/s * 0.561
Vy = 0.4779 m/s

Now that we have the initial vertical velocity, we can use it to determine the time it takes for the spider to reach the peak of its trajectory (when Vy becomes 0). This can be done using the kinematic equation:

Vy = Vy0 + a * t
0 = 0.4779 m/s + (-9.8 m/s^2) * t

Solving for t:
t = -0.4779 m/s / (-9.8 m/s^2)
t ≈ 0.0488 s

Since the spider takes 0.0780 s to travel from the table to the magazine, and approximately half of that time is spent before reaching the peak, we can calculate the time it takes to fall from the peak to the magazine:

Time to fall = Total time - Time to peak
t_fall = 0.0780 s - 0.0488 s
t_fall = 0.0292 s

Using this time, we can find the distance fallen (d_fall) in the vertical direction using the kinematic equation:

d = v * t + (1/2) * a * t^2,
d_fall = 0.4779 m/s * 0.0292 s + (1/2) * (-9.8 m/s^2) * (0.0292 s)^2
d_fall = 0.01398 m

Finally, to find the thickness of the magazine, we equate the vertical distance fallen to the thickness of the magazine:

Thickness = d_fall

Converting the result to millimeters:
Thickness = 0.01398 m * 1000 mm/m
Thickness ≈ 13.98 mm

Therefore, the thickness of the magazine is approximately 13.98 millimeters.