A 2561-kg van runs into the back of a 823-kg compact car at rest. They move off together at 8.5 m/s. Assuming the friction with the road is negligible, calculate the initial speed of the van.
final momentum = initial momentum
(2561+823)(8.5)= 2561 V + 823 (0)
Final momentum = Initial Momentum
(2561+823)(8.5) = 2561V + 823(0)
(3384)(8.5) = 2561V
28764 = 2561V
11.23m/s = V
To solve this problem, we can apply the principle of conservation of momentum. According to this principle, the total momentum before the collision should be equal to the total momentum after the collision.
Let's denote the initial velocity of the van as "v" and the final velocity of both vehicles as 8.5 m/s.
The momentum before the collision can be calculated by multiplying the mass of each object by its initial velocity. Therefore:
Initial momentum of the van = mass of the van * initial velocity of the van
= 2561 kg * v
Initial momentum of the compact car = mass of the compact car * initial velocity of the compact car
= 823 kg * 0 (since the compact car is at rest)
Since the total momentum before the collision is equal to the total momentum after the collision, we can set up the following equation:
Initial momentum of the van + Initial momentum of the compact car = Final momentum of the van + Final momentum of the compact car
2561 kg * v + 0 = 2561 kg * 8.5 m/s + 823 kg * 8.5 m/s
Simplifying the equation:
2561 kg * v = (2561 kg + 823 kg) * 8.5 m/s
2561 kg * v = 3384 kg * 8.5 m/s
2561 kg * v = 28764 kg*m/s
Now, we can divide both sides of the equation by 2561 kg to solve for v:
v = 28764 kg*m/s / 2561 kg
v ≈ 11.24 m/s
Therefore, the initial speed of the van is approximately 11.24 m/s.
To find the initial speed of the van before the collision, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision should be equal to the total momentum after the collision. The formula for momentum is:
Momentum = mass × velocity
Let's assume the initial speed of the van is "v" m/s. Since the compact car is at rest, its initial speed is zero. Therefore, the total momentum before the collision is:
Initial momentum = momentum of van + momentum of compact car
= (mass of van × velocity of van) + (mass of compact car × velocity of compact car)
= (2561 kg × v) + (823 kg × 0)
= 2561v kg⋅m/s
After the collision, the van and compact car stick together and move with a common speed of 8.5 m/s. Therefore, the total momentum after the collision is:
Final momentum = mass of the combined system × final velocity
= (mass of van + mass of compact car) × 8.5 m/s
= (2561 kg + 823 kg) × 8.5 m/s
= 3384 × 8.5 kg⋅m/s
According to the principle of conservation of momentum, the initial momentum and final momentum should be equal. Therefore, we can set up the equation:
Initial momentum = Final momentum
2561v kg⋅m/s = 3384 × 8.5 kg⋅m/s
Now, we can solve for "v":
2561v = 3384 × 8.5
v = (3384 × 8.5) / 2561
v ≈ 11.23 m/s
So, the initial speed of the van before the collision was approximately 11.23 m/s.