A 13,900 kg truck is moving at 23.5 m/s on a mountain road when the brakes are applied. The brakes FAIL!! Fortunately, the driver sees a deceleration (ouch!) ramp a short distance ahead. This is an incline used by out of control trucks to dissipate their KE by converting it to PE in order to stop safely. Since the brakes have failed, you may ignore friction and air resistance.
a) What is the truck’s original kinetic energy?
b) Find the speed of the truck at a point on the ramp that is 12.5 m higher than the point where he entered the ramp.
c) How far above the point where he entered the ramp will the truck be when it comes to rest?
a) KE=mv²/2= 13900•23.5²/2 = 3.83•10⁶ J.
b) KE= PE₁+KE₁
KE₁ = KE-PE = mv²/2 -mgh₁ =
=3.83•10⁶ - 13900•9.8•12.5=
=3.83•10⁶- 1.7•10⁶ =2.13•10⁶ J.
mv₁²/2 = KE₁
v₁ =sqrt{ 2•KE₁/m }= sqrt{2•2.13•10⁶/13900 }=
=17.5 m/s
b)
KE =PE
mv²/2 = mgh
h= mv²/2mg =
=v²/2g = 23.5²/2•9.8 = 28.2 m
To solve this problem, we need to use the principles of kinetic energy and potential energy.
a) To find the truck's original kinetic energy, we can use the formula for kinetic energy:
Kinetic Energy (KE) = 0.5 * mass * velocity^2
Given:
Mass (m) = 13,900 kg
Velocity (v) = 23.5 m/s
Substituting these values into the formula, we get:
KE = 0.5 * 13,900 kg * (23.5 m/s)^2
KE ≈ 0.5 * 13,900 kg * 552.25 m^2/s^2
KE ≈ 3,057,088.75 Joules
Therefore, the truck's original kinetic energy is approximately 3,057,088.75 Joules.
b) To find the speed of the truck at a point on the ramp that is 12.5 m higher than the entry point, we need to use the principle of conservation of energy. At the top of the incline, the truck's kinetic energy will be fully converted to potential energy.
The total mechanical energy of the system remains constant, so:
Kinetic Energy (KE) + Potential Energy (PE1) = Potential Energy (PE2)
Let's define the following variables:
v1 - Speed at the entry point
v2 - Speed at the higher point on the ramp
Since the truck comes to rest at the higher point, its speed is 0, so v2 = 0.
Using the formula for potential energy:
PE = mass * gravitational acceleration * height
The potential energy at the entry point, PE1, can be calculated as:
PE1 = mass * gravitational acceleration * height1
The potential energy at the higher point, PE2, can be calculated as:
PE2 = mass * gravitational acceleration * (height1 + height2)
Since we have the same mass and gravitational acceleration in both equations, we can equate the two potential energy values:
PE1 = PE2
mass * gravitational acceleration * height1 = mass * gravitational acceleration * (height1 + height2)
Canceling out mass and gravitational acceleration, our equation becomes:
height1 = height1 + height2
Simplifying further:
height2 = 0
Therefore, the height at the higher point on the ramp is equal to 0. This means the truck will have zero height and therefore zero velocity when it reaches that point.
c) To find how far above the entry point the truck will be when it comes to rest, we can calculate the change in potential energy using the formula:
Change in Potential Energy (ΔPE) = mass * gravitation acceleration * change in height
The change in height is given as 12.5 m. Therefore, the change in potential energy is:
ΔPE = mass * gravitational acceleration * change in height
ΔPE = 13,900 kg * 9.8 m/s^2 * 12.5 m
ΔPE ≈ 1,699,025 Joules
This change in potential energy represents the energy dissipated to bring the truck to rest. Since all the kinetic energy of the truck is converted to potential energy, this is the same as the truck's original kinetic energy.
Therefore, the truck will be approximately 12.5 meters above the entry point when it comes to rest.