1.A 75 kg passenger in a van is wearing a seat belt whent he van moving at 15m/s, collides with a conrete wall. The front end of the van collapses 0.50 m as it comes to rest.
a) what was the paseenger's kinetic energy before the crash
b) What average force did the seat belt exert on the passengfer during the crash?
2. A 1400 kg car is accelerating up a hill. The hill is 150m long and the total rise of the hill is 6.0m. The car accelerates from a speed of 7.0m/s at the bottom to 15m/s at the top. If the average retarding force of friction is 700 N find the change in potential energy of the car, and the change in kinetic energy of the car.
KE=1/2 m v^2
force*distance=workaborbed=KEinitial
2. change in PE= 1400*g*6m
change in PE=finailKE-initialKE
= 1/2 m (15^2-7^2)
1a) To find the passenger's kinetic energy before the crash, we can use the formula for kinetic energy:
Kinetic Energy (KE) = (1/2) * mass * velocity^2
Given:
- Mass (m) = 75 kg
- Velocity (v) = 15 m/s
Substituting the values into the formula, we have:
KE = (1/2) * 75 kg * (15 m/s)^2
KE = 1/2 * 75 kg * 225 m^2/s^2
KE = 1/2 * 16875 kg * m^2/s^2
KE = 8437.5 kg * m^2/s^2 (or J, Joules)
Therefore, the passenger's kinetic energy before the crash was 8437.5 Joules.
1b) To find the average force exerted by the seat belt, we can use the work-energy principle:
Work (W) = Force (F) * distance (d)
Since the work done in stopping the passenger is equal to the change in kinetic energy, we can write:
Force * distance = Change in kinetic energy
F * 0.5 m = KE (final) - KE (initial)
F * 0.5 m = 0 - 8437.5 J
F * 0.5 m = -8437.5 J (note that the change in kinetic energy is negative because it went from a positive value to zero)
We can now solve for the average force:
F = -8437.5 J / 0.5 m
F = -16875 N
The negative sign indicates that the force exerted by the seat belt is directed opposite to the motion, providing the necessary deceleration.
Therefore, the average force exerted by the seat belt on the passenger during the crash was 16875 Newtons.
To solve these problems, we'll need to use the formulas for kinetic energy (KE), work (W), average force (F_avg), change in potential energy (ΔPE), and change in kinetic energy (ΔKE). Here are the formulas we'll be using:
1. Kinetic Energy (KE) = (1/2)mv^2
2. Work (W) = F*d*cosθ (θ is the angle between the force and displacement)
3. Average Force (F_avg) = W/d
4. Change in Potential Energy (ΔPE) = mgh (h is the change in height)
5. Change in Kinetic Energy (ΔKE) = KE_final - KE_initial
Now let's solve the problems step by step:
1. a) What was the passenger's kinetic energy (KE) before the crash?
Using the formula KE = (1/2)mv^2, where m is the mass and v is the velocity, we can calculate the passenger's kinetic energy.
KE = (1/2)(75 kg)(15 m/s)^2
KE = (1/2)(75 kg)(225 m^2/s^2)
KE = 5062.5 Joules
Therefore, the passenger's kinetic energy before the crash was 5062.5 Joules.
1. b) What average force did the seat belt exert on the passenger during the crash?
We can use the work formula, W = F*d*cosθ, to find the work done by the average force exerted by the seat belt.
Given that the front end of the van collapses 0.50 m as it comes to rest, we can assume d = 0.50 m.
Since the work done is equal to the change in kinetic energy, W = ΔKE, we can rearrange the formula to find the average force:
F_avg = W/d
Now substitute the values:
F_avg = ΔKE / d
F_avg = KE_final - KE_initial / d
We've already calculated the initial kinetic energy as 5062.5 Joules. To find the final kinetic energy, we need to calculate the work done by the collision in terms of joules.
The work done by the collision is given by W = F*d*cosθ. In this case, the work is equal to the change in kinetic energy, so W = ΔKE = KE_final - KE_initial.
Since the van comes to rest, the final kinetic energy (KE_final) is 0.
Therefore, W = ΔKE = KE_final - KE_initial = 0 - 5062.5 = -5062.5 Joules
Now, plug in the values:
F_avg = -5062.5 J / 0.50 m
F_avg = -10125 N
The average force exerted by the seat belt on the passenger during the crash was 10125 Newtons (opposite in direction to the motion).
2. To solve this problem, follow the steps:
a) Change in Potential Energy (ΔPE):
Using the formula ΔPE = mgh, where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the change in height, we can calculate the change in potential energy.
ΔPE = (1400 kg)(9.8 m/s^2)(6.0 m)
ΔPE = 82320 Joules
Therefore, the change in potential energy of the car is 82320 Joules.
b) Change in Kinetic Energy (ΔKE):
Using the formula ΔKE = KE_final - KE_initial, we can calculate the change in kinetic energy.
The initial kinetic energy (KE_initial) can be calculated using the formula KE = (1/2)mv^2:
KE_initial = (1/2)(1400 kg)(7.0 m/s)^2
KE_initial = 34300 Joules
The final kinetic energy (KE_final) can be calculated using the formula KE = (1/2)mv^2:
KE_final = (1/2)(1400 kg)(15 m/s)^2
KE_final = 157500 Joules
Now, calculate the change in kinetic energy:
ΔKE = KE_final - KE_initial
ΔKE = 157500 J - 34300 J
ΔKE = 123200 Joules
Therefore, the change in kinetic energy of the car is 123200 Joules.