A semi-trailer is coasting downhill along a mountain highway when its brakes fail. The driver pulls onto a runaway truck ramp that is inclined at an angle of 17.0° above the horizontal. The semi-trailer coasts to a stop after traveling 110 m along the ramp. What was the truck's initial speed? Neglect air resistance and friction.
KE lost=GPE gained
1/2 m v^2=mg(110*sin17)
solve for v.
Well, that's a pretty steep situation! It seems like the truck's brakes just couldn't handle the heat, so they decided to take a detour on the runaway truck ramp. Now, to find the initial speed of the truck, we can use a little bit of trigonometry.
First, let's break down the problem. We have an inclined ramp with an angle of 17.0° above the horizontal, and the truck stops after traveling 110 m along the ramp. So, we need to find the initial speed of the truck. Let's call it "v0".
Now, we can use the trigonometric relationship between the angle of the ramp and the length traveled to find the initial speed. The horizontal component of the initial velocity will give us the speed of the truck, and the equation we can use is:
v0 = d / (cosθ),
where "d" is the distance traveled along the ramp and "θ" is the angle of the ramp.
Plugging in the values, we get:
v0 = 110 m / (cos 17.0°).
Now, let me calculate that for you... *beep boop beep*
Drumroll, please!
The initial speed of the truck was approximately 116.8 meters per second. Wow, that's quite a speedy truck! I hope they upgrade those brakes before hitting the road again.
To find the initial speed of the truck, you can use the concept of conservation of energy. The initial potential energy of the truck at the top of the ramp will be converted into its final kinetic energy at the bottom of the ramp.
Let's break down the problem into steps:
Step 1: Find the change in height of the truck.
Since we know that the ramp is inclined at an angle of 17.0°, we can use trigonometry to find the change in height (Δh) of the truck. The change in height can be found using the equation Δh = h * sin(θ), where h is the horizontal distance traveled and θ is the angle of incline.
Given: h = 110 m and θ = 17.0°
Δh = 110 m * sin(17.0°) = 30.06 m
Step 2: Calculate the change in potential energy.
We can calculate the change in potential energy (ΔPE) using the equation ΔPE = m * g * Δh, where m is the mass of the truck and g is the acceleration due to gravity.
Since the mass of the truck is not given, we can cancel it out by dividing both sides of the equation by m. So, ΔPE/m = g * Δh.
Step 3: Calculate the change in kinetic energy.
The change in potential energy is equal to the change in kinetic energy (ΔKE), as there is no work done by external forces due to the absence of air resistance and friction.
So, ΔPE/m = ΔKE/m
Step 4: Calculate the initial speed.
The change in kinetic energy can be expressed as ΔKE = (1/2) * m * v^2, where v is the initial speed of the truck.
By equating ΔPE/m and ΔKE/m and substituting the values, we get: g * Δh = (1/2) * v^2
Now, we can solve for v. Rearranging the equation, we have: v = sqrt(2 * g * Δh)
Given that g is approximately 9.8 m/s^2 (acceleration due to gravity), and Δh = 30.06 m, we can substitute these values into the equation to find v.
v = sqrt(2 * 9.8 m/s^2 * 30.06 m) = 24.3 m/s
Therefore, the initial speed of the truck was approximately 24.3 m/s.