A particle travels horizontally between two
parallel walls separated by 18.4 m. It moves
toward the opposing wall at a constant rate
of 5.9 m/s. Also, it has an acceleration in the
direction parallel to the walls of 2.2 m/s^2. a) What will be its speed when it hits the opposing wall? Answer in units of m/s. b) At what angle with the wall will the particle strike? Answer in units of °.
time to hit = 18.4/5.9 = 3.12s
(a) = √(18.4^2 + (2.2*3.12)^2) =
(b) arctan(18.4/6,86)
To solve this problem, we can use the kinematic equations of motion. We'll break down the problem into two parts: finding the time it takes for the particle to hit the wall and then using that time to find the speed and angle of impact.
a) To determine the time it takes for the particle to hit the wall, we can use the equation:
s = ut + (1/2)at^2
where
s = distance between the walls = 18.4 m
u = initial velocity = 5.9 m/s
a = acceleration = 2.2 m/s^2
t = time taken to hit the wall (unknown)
Rearranging the equation, we get:
18.4 = (5.9)t + (1/2)(2.2)(t^2)
Simplifying the equation, we have:
1.1t^2 + 5.9t - 18.4 = 0
Now, we can solve this quadratic equation using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values, we get:
t = [(-5.9) ± √((5.9)^2 - 4(1.1)(-18.4))] / (2(1.1))
Solving this equation, we find two values for t: t1 ≈ 1.655 s and t2 ≈ -3.544 s. Since time cannot be negative, we discard t2.
Therefore, the time it takes for the particle to hit the opposite wall is approximately 1.655 seconds.
b) To find the speed when the particle hits the wall, we can use the equation:
v = u + at
where
v = final velocity (unknown)
u = initial velocity = 5.9 m/s
a = acceleration = 2.2 m/s^2
t = time taken to hit the wall = 1.655 s
Plugging in the values, we have:
v = 5.9 + (2.2)(1.655)
Calculating this, we find that the speed when the particle hits the opposing wall is approximately 9.219 m/s.
c) To determine the angle at which the particle strikes the wall, we can use trigonometry. The particle's horizontal displacement is given by the distance between the walls, which is 18.4 m.
The vertical displacement can be calculated using the equation:
h = (1/2)gt^2
where
h = vertical displacement (unknown)
g = acceleration due to gravity = 9.8 m/s^2
t = time taken to hit the wall = 1.655 s
Plugging in the values, we have:
h = (1/2)(9.8)(1.655)^2
Simplifying this equation, we find that the vertical displacement is approximately 13.527 m.
Now, we can use the tangent function to find the angle:
tan(θ) = h / s
where
θ = angle with the wall (unknown)
h = vertical displacement = 13.527 m
s = horizontal displacement = 18.4 m
Plugging in the values, we have:
tan(θ) = 13.527 / 18.4
Calculating this, we find that the angle at which the particle strikes the wall is approximately 36.95°.
So, the answers to the given questions are:
a) The speed when it hits the opposing wall is approximately 9.219 m/s.
b) The angle with the wall at which the particle strikes is approximately 36.95°.