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?
To answer these questions, we need to use the concepts of kinetic energy (KE), potential energy (PE), and the conservation of mechanical energy. Let's break down each question step by step:
a) What is the truck's original kinetic energy?
The formula for kinetic energy is KE = (1/2) * mass * velocity^2. Given the mass of the truck, which is 13,900 kg, and its velocity, which is 23.5 m/s, we can plug in these values to calculate the kinetic energy.
KE = (1/2) * 13,900 kg * (23.5 m/s)^2
Simplifying the equation:
KE = 0.5 * 13,900 kg * (23.5 m/s)^2
Calculating the value:
KE = 0.5 * 13,900 kg * 551.75 m^2/s^2
KE = 3,823,987.5 Joules
Therefore, the truck's original kinetic energy is 3,823,987.5 Joules.
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.
Since the brakes have failed, the truck's kinetic energy will be converted to potential energy as it moves up the ramp. In this scenario, the total mechanical energy is conserved because we are ignoring friction and air resistance. Therefore, at any point on the ramp, the sum of the truck's kinetic and potential energy will be equal to its original kinetic energy.
Let's assume the point where the truck entered the ramp as the reference level for potential energy (PE = 0). At a point 12.5 m higher, the potential energy gained will be mgh, where m is the mass of the truck, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height gain (12.5 m).
Using the conservation of mechanical energy:
Original KE = KE + PE
3,823,987.5 Joules = KE + mgh
We need to find the value of KE when the truck is 12.5 m higher, so we can rearrange the equation as follows:
KE = 3,823,987.5 Joules - mgh
Plugging in the known values:
KE = 3,823,987.5 Joules - (13,900 kg * 9.8 m/s^2 * 12.5 m)
Calculating the value:
KE = 3,823,987.5 Joules - 1,932,250 Joules
KE = 1,891,737.5 Joules
Now, we can calculate the speed (v) of the truck at this point using the kinetic energy formula:
KE = (1/2) * m * v^2
Plugging in the values:
1,891,737.5 Joules = (1/2) * 13,900 kg * v^2
Rearranging the equation:
v^2 = (2 * 1,891,737.5 Joules) / 13,900 kg
Calculating the value:
v^2 = 271.02 m^2/s^2
Taking the square root of both sides:
v = √(271.02 m^2/s^2)
v ≈ 16.45 m/s
Therefore, the speed of the truck at a point on the ramp 12.5 m higher is approximately 16.45 m/s.
c) How far above the point where he entered the ramp will the truck be when it comes to rest?
When the truck comes to rest, its final kinetic energy will be zero, and all of its initial kinetic energy will be converted to potential energy at the highest point on the ramp.
mgh = Initial KE
13,900 kg * 9.8 m/s^2 * h = 3,823,987.5 Joules
Simplifying the equation:
h = (3,823,987.5 Joules) / (13,900 kg * 9.8 m/s^2)
Calculating the value:
h ≈ 27.95 m
Therefore, the truck will be approximately 27.95 m above the point where it entered the ramp when it comes to rest.
To solve this problem, we will use the principle of conservation of mechanical energy. The total mechanical energy of the truck, which includes its kinetic energy (KE) and potential energy (PE), will remain constant throughout the motion.
a) The truck's original kinetic energy can be calculated using the formula:
KE = (1/2) * mass * velocity^2
Using the given values:
mass = 13,900 kg
velocity = 23.5 m/s
KE = (1/2) * 13,900 kg * (23.5 m/s)^2
KE = 1/2 * 13,900 kg * 552.25 m^2/s^2
KE = 3,093,887.5 Joules
Therefore, the truck's original kinetic energy is 3,093,887.5 Joules.
b) To find the speed of the truck at a point 12.5 m higher on the ramp, we can use the conservation of mechanical energy again. We will equate the initial kinetic energy (KE) to the final potential energy (PE) and kinetic energy (KE).
KE_initial = KE_final + PE_final
Since there is no change in the truck's mass, the kinetic energy formula remains the same. However, the potential energy formula will be different:
PE = mass * gravity * height
Using the given values:
mass = 13,900 kg
gravity = 9.8 m/s^2
height = 12.5 m
KE_initial = KE_final + PE_final
(1/2) * 13,900 kg * (23.5 m/s)^2 = KE_final + (13,900 kg * 9.8 m/s^2 * 12.5 m)
Simplifying the equation:
3656122.5 = KE_final + 1707375
KE_final = 3656122.5 - 1707375
KE_final = 1948747.5 Joules
Now, we can solve for the final velocity (v_final) using the formula:
KE_final = (1/2) * mass * v_final^2
1948747.5 = (1/2) * 13,900 kg * v_final^2
v_final^2 = (1948747.5 * 2) / (13,900 kg)
v_final^2 = 281.015 m^2/s^2
v_final = √(281.015 m^2/s^2)
v_final ≈ 16.77 m/s
Therefore, the speed of the truck at a point 12.5 m higher on the ramp is approximately 16.77 m/s.
c) To find the distance above the point where the truck entered the ramp when it comes to rest, we can use the conservation of mechanical energy again. Equating the final potential energy (PE) to zero (as the truck comes to rest), we can solve for the height (h).
PE = mass * gravity * height
Using the given values:
mass = 13,900 kg
gravity = 9.8 m/s^2
0 = 13,900 kg * 9.8 m/s^2 * height
height = 0
Therefore, the truck will come to rest at the same height as the point where it entered the ramp.