A conveyor belt is used to move sand from one place to another in a factory. The conveyor is tilted at an angle of 18° above the
horizontal and the sand is moved without slipping at the rate of 2 m/s. The sand is collected in a big drum 5 m below the end of the conveyor belt. Determine the horizontal distance between the end of the conveyor belt and the middle of the collecting drum.
1 answer:
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(a) Zero
When the ball reaches its highest point, the direction of motion of the ball reverses (from upward to downward). This means that the velocity is changing sign: this also means that at that moment, the velocity must be zero.
This can be also understood in terms of conservation of energy: when the ball is tossed up, initially it has kinetic energy
where m is the ball's mass and v is the initial speed. As it goes up, this kinetic energy is converted into potential energy, and when the ball reaches the highest point, all the kinetic energy has been converted into potential energy:
where g is the gravitational acceleration and h is the height of the ball at highest point. At that point, therefore, the potential energy is maximum, while the kinetic energy is zero, and so the velocity is also zero.
(b) 9.8 m/s upward
We can find the velocity of the ball 1 s before reaching its highest point by using the equation:
where
a = g = -9.8 m/s^2 is the acceleration due to gravity, which is negative since it points downward
v = 0 is the final velocity (at the highest point)
u is the initial velocity
t = 1 s is the time interval
Solving for u, we find
and the positive sign means it points upward.
(c) -9.8 m/s
The change in velocity during the 1-s interval is given by
where
v = 0 is the final velocity (at the highest point)
u = 9.8 m/s is the initial velocity
Substituting, we find
(d) 9.8 m/s downward
We can find the velocity of the ball 1 s after reaching its highest point by using again the equation:
where this time we have
a = g = -9.8 m/s^2 is the acceleration due to gravity, still negative
v is the final velocity (1 s after reaching the highest point)
u = 0 is the initial velocity (at the highest point)
t = 1 s is the time interval
Solving for v, we find
and the negative sign means it points downward.
(e) -9.8 m/s
The change in velocity during the 1-s interval is given by
where here we have
v = -9.8 m/s is the final velocity (1 s after reaching the highest point)
u = 0 is the initial velocity (at the highest point)
Substituting, we find
(f) -19.6 m/s
The change in velocity during the overall 2-s interval is given by
where in this case we have:
v = -9.8 m/s is the final velocity (1 s after reaching the highest point)
u = +9.8 m/s is the initial velocity (1 s before reaching the highest point)
Substituting, we find
(g) -9.8 m/s^2
There is always one force acting on the ball during the motion: the force of gravity, which is given by
where
m is the mass of the ball
g = -9.8 m/s^2 is the acceleration due to gravity
According to Newton's second law, the resultant of the forces acting on the body is equal to the product of mass and acceleration (a), so
which means that the acceleration is
and the negative sign means it points downward.
Answer:
block velocity v = 0.09186 = 9.18 10⁻² m/s and speed bollet v₀ = 11.5 m / s
Explanation:
We will solve this problem using the concepts of the moment, let's try a system formed by the two bodies, the bullet and the block; In this system all scaffolds during the crash are internal, consequently, the moment is preserved.
Let's write the moment in two moments before the crash and after the crash, let's call the mass of the bullet (m) and the mass of the Block (M)
Before the crash
p₀ = m v₀ + 0
After the crash
= (m + M) v
p₀ =
m v₀ = (m + M) v (1)
Now let's lock after the two bodies are joined, in this case the mechanical energy is conserved, write it in two moments after the crash and when you have the maximum compression of the spring
Initial
Em₀ = K = ½ m v2
Final
E = Ke = ½ k x2
Emo = E
½ m v² = ½ k x²
v² = k/m x²
Let's look for the spring constant (k), with Hook's law
F = -k x
k = -F / x
k = - 0.75 / -0.25
k = 3 N / m
Let's calculate the speed
v = √(k/m) x
v = √ (3/8.00) 0.15
v = 0.09186 = 9.18 10⁻² m/s
This is the spped of the block plus bullet rsystem right after the crash
We substitute calculate in equation (1)
m v₀ = (m + M) v
v₀ = v (m + M) / m
v₀ = 0.09186 (0.008 + 0.992) /0.008
v₀ = 11.5 m / s