A runaway truck lane heads uphill at 30◦ to the horizontal.
If an out of control 15000 kg truck enters the lane going at 30 m/s , how far along the ramp does it go? The acceleration due to gravity is 10 m/s2.
Distance = (30 m/s)^2 / (2 * 10 m/s2 * sin(30◦))
Distance = 81.65 m
Why did the truck decide to go for a hill joyride? It must have been tired of being stuck in traffic! Well, let's calculate how far it managed to go uphill.
We know the angle of the ramp, which is 30 degrees. To make things easier, let's decompose the weight of the truck into two components: one acting perpendicular to the ramp and the other parallel to the ramp.
The component of the weight perpendicular to the ramp is given by:
W_perpendicular = m * g * cos(theta)
where m is the mass of the truck (15000 kg), g is the acceleration due to gravity (10 m/s^2), and theta is the angle of the ramp (30 degrees).
Let's calculate that:
W_perpendicular = 15000 kg * 10 m/s^2 * cos(30 degrees)
Once we have the perpendicular component, we can calculate the force parallel to the ramp required to stop the truck using Newton's second law:
F_parallel = m * a
where m is the mass of the truck (15000 kg) and a is the acceleration parallel to the ramp.
Now, the force parallel to the ramp is the difference between the component of the weight parallel to the ramp and the force due to the truck's initial velocity:
F_parallel = W_parallel - F_initial
where F_initial is the force due to the truck's initial velocity (m * v) and W_parallel is the component of the weight parallel to the ramp:
F_parallel = W_parallel - m * v
Given that the truck's initial velocity is 30 m/s and m = 15000 kg, we can calculate the force:
F_parallel = W_parallel - (15000 kg * 30 m/s)
Now, using the relation F_parallel = m * a, we can find the acceleration parallel to the ramp:
a = F_parallel / m
Finally, to calculate the distance the truck travels uphill, we can use the kinematic equation:
s = (v^2 - u^2) / (2 * a)
where v is the final velocity (which is 0 m/s since the truck comes to a stop), u is the initial velocity (30 m/s), and a is the acceleration calculated earlier.
With all these calculations, we can determine how far the runaway truck traveled uphill. But remember, even trucks need to obey traffic laws and not go on joyrides!
To determine how far the truck goes along the ramp, we need to find the distance traveled in the horizontal direction (x-direction) and the distance traveled in the vertical direction (y-direction).
First, let's find the vertical distance (y-direction) traveled by using the time it takes for the truck to reach the end of the ramp.
1. Resolve the truck's initial velocity (u) into its horizontal (u_x) and vertical (u_y) components.
u_x = u * cosθ
u_x = 30 m/s * cos(30°)
u_x = 30 m/s * √(3)/2
u_x ≈ 25.98 m/s
u_y = u * sinθ
u_y = 30 m/s * sin(30°)
u_y = 30 m/s * 1/2
u_y = 15 m/s
2. Calculate the time (t) it takes for the truck to reach the end of the ramp using the vertical component of the initial velocity and the acceleration due to gravity.
v_y = u_y + a_y * t
0 = 15 m/s + (-10 m/s^2) * t
Solving for t:
10 m/s^2 * t = 15 m/s
t = 15 m/s / 10 m/s^2
t = 1.5 s
Now, let's find the horizontal distance (x-direction) traveled during this time.
3. Calculate the horizontal distance (x) traveled using the horizontal component of the initial velocity and the time taken.
x = u_x * t
x = 25.98 m/s * 1.5 s
x ≈ 38.97 m
Therefore, the truck travels approximately 38.97 meters along the ramp.
To find the distance along the ramp that the runaway truck travels, we can use the principles of projectile motion.
First, let's consider the components of the truck's initial velocity. The velocity can be broken down into two components: one along the ramp and one perpendicular to it.
The component of the initial velocity along the ramp can be calculated using trigonometry. Given that the truck enters the lane at an angle of 30 degrees with the horizontal and has an initial velocity of 30 m/s, we can calculate the initial velocity along the ramp as follows:
Initial velocity along the ramp = initial velocity * sin(angle)
= 30 m/s * sin(30°)
= 15 m/s
Next, we need to find the time it takes for the truck to stop moving along the ramp. Since the truck experiences a deceleration due to gravity along the ramp, we can use the kinematic equation:
Final velocity = Initial velocity + (acceleration * time)
Here, the final velocity along the ramp is 0 m/s since the truck comes to a stop. The acceleration due to gravity is 10 m/s^2, and the initial velocity along the ramp is 15 m/s. By substituting these values into the equation, we can find the time taken for the truck to stop:
0 m/s = 15 m/s + (-10 m/s^2 * time)
10 m/s^2 * time = 15 m/s
time = 15 m/s / 10 m/s^2
time = 1.5 s
Now that we know the time taken for the truck to stop, we can find the distance traveled along the ramp. We can use the kinematic equation:
Distance = Initial velocity * time + (1/2) * acceleration * time^2
Substituting the known values into the equation:
Distance = 15 m/s * 1.5 s + (1/2) * (-10 m/s^2) * (1.5 s)^2
Distance = 22.5 m + (1/2) * (-10 m/s^2) * 2.25 s^2
Distance = 22.5 m - 11.25 m
Distance = 11.25 m
Therefore, the truck travels a distance of 11.25 meters along the ramp.