A blue car with mass mc = 528 kg is moving east with a speed of vc = 17 m/s and collides with a purple truck with mass mt = 1352 kg that is moving south with an unknown speed. The two collide and lock together after the collision moving at an angle of θ = 59° South of East
a. What is the magnitude of the initial momentum of the car?
b. What is the magnitude of the initial momentum of the truck?
c. What is the speed of the truck before the collision?
d. What is the magnitude of the momentum of the car-truck combination immediately after the collision?
e. What is the speed of the car-truck combination immediately after the collision?
a. momentum of car (Pc) = mass*velocity
momentum = (528kg)*(17m/s)
= 8976 kg*m/s
Note: we need to find part c first
b. momentum of truck (Pt) = mass*velocity
momentum = (1352kg)*(11m/s)
= 14872 kg*m/s
c. tan(θ) = (momentum truck/momentum car)
tan(59°) = (1352*v)/(8976)
solve for v
v = 11 m/s
d. total momentum = sqrt(Pc^2 +Pt^2)
total momentum = sqrt((8976^2)+(14872^2))
= 17370.81 kg*m/s
e. final velocity = (total momentum)/(total mass)
= 17370.81/(528+1352)
= 9.24 m/s
Note: The magnitude of the TOTAL momentum of the system before and after the collision will be the same.
To solve this problem, we can use the conservation of momentum principle. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
a. To find the magnitude of the initial momentum of the car, we can use the formula:
Initial momentum of the car = mass of the car * velocity of the car
Given that the mass of the car (mc) is 528 kg and the velocity of the car (vc) is 17 m/s, we can calculate:
Initial momentum of the car = 528 kg * 17 m/s = 8976 kg·m/s
b. To find the magnitude of the initial momentum of the truck, we can use the same formula:
Initial momentum of the truck = mass of the truck * velocity of the truck
However, the velocity of the truck is unknown in this problem. We'll need to calculate it using the angle and speed information.
c. To find the speed of the truck before the collision, we'll use trigonometry. The given angle is 59° South of East, which means the truck is moving at an angle of 180° - 59° = 121° with respect to the east direction.
Using the trigonometric relationships, we can determine the truck's velocity in the x and y directions:
Velocity of the truck in the x direction (vx) = speed of the truck * cos(angle)
Velocity of the truck in the y direction (vy) = speed of the truck * sin(angle)
Since the truck is moving south, vy is negative. Given that the angle is 121°, we have:
vx = speed of the truck * cos(121°)
vy = -speed of the truck * sin(121°)
The magnitude of the truck's velocity is given by the Pythagorean theorem:
Speed of the truck = sqrt(vx^2 + vy^2)
We can calculate the speed of the truck using these equations.
d. To find the magnitude of the momentum of the car-truck combination immediately after the collision, we need to add their momenta together.
Momentum of the car-truck combination = momentum of the car + momentum of the truck
e. To find the speed of the car-truck combination immediately after the collision, we divide the momentum of the car-truck combination by their combined mass.
Speed of the car-truck combination = (momentum of the car + momentum of the truck) / (mass of the car + mass of the truck)
Now we can proceed with the calculations.
To solve this problem, we will use the conservation of momentum principle, which states that the total momentum before a collision is equal to the total momentum after the collision.
a. To find the magnitude of the initial momentum of the car, we can use the formula: momentum = mass × velocity. Thus, the initial momentum of the car (Pc) is given by:
Pc = mc × vc
Substituting the given values:
Pc = 528 kg × 17 m/s = 8,976 kg·m/s
Therefore, the magnitude of the initial momentum of the car is 8,976 kg·m/s.
b. Similarly, to find the magnitude of the initial momentum of the truck (Pt), we can use the same formula:
Pt = mt × vt
However, the velocity of the truck (vt) is unknown. Let's denote it as vt and its angle of motion as θt.
c. To find the speed of the truck before the collision, we can use trigonometric relationships. The velocity of the truck can be represented as:
vt = vtsin(θt) (since it is moving south)
We are given that the angle θ is 59° South of East. To determine θt, we subtract 180° (due South) from 59°:
θt = 59° - 180° = -121°
However, since it is moving south, θt should be a positive angle. We can convert -121° to its positive equivalent by adding 180°:
θt = -121° + 180° = 59°
Now, we can use the given angle and the velocity of the truck after the collision (which is locked together with the car) to find the velocity of the truck before the collision.
Using the trigonometric relationship, we have:
vt = (vcos(θ) + vtsin(θ)) / cos(θt)
Substituting the given values:
17 m/s = (vcos(59°) + vt × sin(59°)) / cos(59°)
Simplifying the equation and solving for vt:
vt × cos(59°) = 17 m/s - vcos(59°)
vt = (17 m/s - vcos(59°)) / cos(59°)
Now, substitute the value of cos(59°) and calculate vt:
vt = (17 m/s - 8.5 m/s) / 0.5
vt = 8.5 m/s / 0.5
vt = 17 m/s
Thus, the speed of the truck before the collision is 17 m/s.
d. Since momentum is conserved, the magnitude of the momentum of the car-truck combination immediately after the collision (Pf) is equal to the initial momentum, Pc + Pt:
Pf = Pc + Pt
Substituting the values:
Pf = 8,976 kg·m/s + (1352 kg × 17 m/s) = 8,976 kg·m/s + 22,984 kg·m/s = 31,960 kg·m/s
Therefore, the magnitude of the momentum of the car-truck combination immediately after the collision is 31,960 kg·m/s.
e. To find the speed of the car-truck combination immediately after the collision, we need to divide the momentum by the total mass of the combined system.
Total mass (mtotal) = mc + mt = 528 kg + 1352 kg = 1880 kg
Speed (vf) = Pf / mtotal
Substituting the values:
vf = 31,960 kg·m/s / 1880 kg = 17 m/s
Hence, the speed of the car-truck combination immediately after the collision is 17 m/s.