A spider crawling across a table leaps onto a magazine blocking its path. The initial velocity of the spider is 0.890 m/s at an angle of 33.6° above the table, and it lands on the magazine 0.0750 s after leaving the table. Ignore air resistance. How thick is the magazine? Express your answer in millimeters.
Vo = 0.890 m/s @ 33.6 Deg.
Xo = Hor. = 0.890*cos33.6 = 0.741 m/s.
Yo = Ver. = 0.890*sin33.6 = 0.493 m/s.
Tr = (Yf-Yo) / g,
Tr = (0-0.493) / -9.8 = 0.0503 s. = Rise time or time to reach max. ht.
hmax = (Yf^2-Yo^2) / 2g,
hmax = (0-(0.493)^2) / -19.6=0.0124 m.
Tr + Tf = 0.0750 s.
0.0503 + Tf = 0.0750,
Tf = 0.0247 s. = Fall time.
d = Vo*t + 0.5g*t*t^2,
d = 0 + 4.9(0.0247)^2 = 0.0030 m.=Fall
distance.
hmax - d = 0.0124 - 0.003=9.4*10^-3 m
= 9.4 mm. = Thickness of mag.
To find the thickness of the magazine, we can start by analyzing the motion of the spider.
First, let's break down the initial velocity of the spider into its horizontal and vertical components.
The horizontal component of the velocity (Vx) remains constant throughout the motion and can be found using the equation:
Vx = V * cos(θ)
where V is the initial velocity of the spider (0.890 m/s) and θ is the angle above the table (33.6°).
Vx = 0.890 m/s * cos(33.6°)
Vx = 0.890 m/s * 0.832
Vx = 0.739088 m/s
The vertical component of the velocity (Vy) changes due to the acceleration due to gravity. The formula to find the time it takes for the spider to reach the magazine (t) is:
t = (2 * Vy) / g
where g is the acceleration due to gravity near the surface of the Earth (9.8 m/s^2).
Substitute the given values into the equation:
0.0750 s = (2 * Vy) / 9.8 m/s^2
Simplifying the equation gives:
Vy = (0.0750 s * 9.8 m/s^2) / 2
Vy = 0.3675 m/s
Since the spider lands on the magazine, the vertical displacement (Δy) can be calculated using the equation:
Δy = Vy * t + (0.5 * g * t^2)
where Δy is the displacement of the spider in the y-direction.
Since the spider leaps off the table and lands on the magazine, the displacement in the y-direction (Δy) is equal to the thickness of the magazine.
Using the given values:
Δy = 0.3675 m/s * 0.0750 s + (0.5 * 9.8 m/s^2 * (0.0750 s)^2)
Δy = 0.02756 m
Finally, we convert the thickness from meters to millimeters:
Thickness (in millimeters) = Δy * 1000
Thickness = 0.02756 m * 1000
Thickness = 27.56 mm
Therefore, the thickness of the magazine is approximately 27.56 millimeters.