A 60 kg boy on roller skates moves with a speed of 5 m/s. He runs into a 40 kg girl on skates. Assuming they cling together after the collision, what is their speed?How much kinetic energy was lost in the collision?
Well, when the boy and girl run into each other, I hope it doesn't turn into a roller derby! Anyway, let's calculate their speed after the collision.
Using the law of conservation of momentum, we can say that the initial momentum is equal to the final momentum. The initial momentum is given by the boy's mass (60 kg) times his initial speed (5 m/s), and the final momentum is given by the combined mass of the boy and girl (60 kg + 40 kg) times their final speed (let's call it V).
So we have:
Initial momentum = Final momentum
(60 kg * 5 m/s) = ((60 kg + 40 kg) * V)
Simplifying that equation gives us:
300 kg * m/s = 100 kg * V
Dividing both sides by 100 kg gives us:
3 m/s = V
So after the collision, their speed is 3 m/s. They slowed down a bit, but at least they're together now!
Now let's calculate the kinetic energy lost in the collision. Kinetic energy is given by the formula KE = 0.5 * mass * velocity squared. Before the collision, the boy's kinetic energy is:
KE_boy = 0.5 * 60 kg * (5 m/s)^2
And the girl's kinetic energy is:
KE_girl = 0.5 * 40 kg * (0 m/s)^2
After the collision, their combined kinetic energy is:
KE_combined = 0.5 * (60 kg + 40 kg) * (3 m/s)^2
Now, let's calculate the kinetic energy lost:
Kinetic energy lost = KE_boy + KE_girl - KE_combined
Put those numbers in and calculate it. I'll wait here...
To find the speed of the boy and girl after the collision, we can use the principle of conservation of linear momentum.
The formula for linear momentum is:
p = m * v
Where p is the momentum, m is the mass, and v is the velocity.
Given that the boy has a mass (m1) of 60 kg and a velocity (v1) of 5 m/s, and the girl has a mass (m2) of 40 kg, we can calculate the total momentum before the collision (p_total_before):
p_total_before = m1 * v1
p_total_before = 60 kg * 5 m/s
p_total_before = 300 kg*m/s
Since momentum is conserved, the total momentum after the collision (p_total_after) is equal to the total momentum before the collision:
p_total_after = p_total_before
Now, let's assume the velocity of the boy and girl after the collision is v_after.
The total momentum after the collision can be calculated as:
p_total_after = (m1 + m2) * v_after
Since we know that the total momentum after the collision is equal to the total momentum before the collision, we can set up an equation:
p_total_before = p_total_after
m1 * v1 = (m1 + m2) * v_after
Substituting the given values:
60 kg * 5 m/s = (60 kg + 40 kg) * v_after
300 kg*m/s = 100 kg * v_after
Dividing both sides by 100 kg:
3 m/s = v_after
Therefore, the speed of the boy and girl after the collision is 3 m/s.
To calculate the kinetic energy lost in the collision, we can use the formula:
Kinetic energy (KE) = 1/2 * m * v^2
The kinetic energy before the collision (KE_before) is:
KE_before = 1/2 * m1 * v1^2
KE_before = 1/2 * 60 kg * (5 m/s)^2
KE_before = 1/2 * 60 kg * 25 m^2/s^2
KE_before = 750 J
The kinetic energy after the collision (KE_after) is:
KE_after = 1/2 * (m1 + m2) * v_after^2
KE_after = 1/2 * 100 kg * (3 m/s)^2
KE_after = 1/2 * 100 kg * 9 m^2/s^2
KE_after = 450 J
The kinetic energy lost in the collision (KE_lost) is:
KE_lost = KE_before - KE_after
KE_lost = 750 J - 450 J
KE_lost = 300 J
Therefore, the kinetic energy lost in the collision is 300 J.
To find their final speed after the collision, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision will be equal to the total momentum after the collision.
Before the collision, the momentum of the boy can be calculated by multiplying his mass (60 kg) by his initial speed (5 m/s). So his momentum is given by:
Momentum of boy = mass of boy * initial speed of boy
= 60 kg * 5 m/s
= 300 kg*m/s
Similarly, the momentum of the girl can be calculated by multiplying her mass (40 kg) by her initial speed. We are assuming that the girl is initially at rest, so her initial speed is zero. Therefore, her momentum is:
Momentum of girl = mass of girl * initial speed of girl
= 40 kg * 0 m/s
= 0 kg*m/s
Total momentum before the collision = momentum of boy + momentum of girl
= 300 kg*m/s + 0 kg*m/s
= 300 kg*m/s
Since the total momentum before the collision is equal to the total momentum after the collision (due to the conservation of momentum), we can set up the equation as follows:
Total momentum before the collision = Total momentum after the collision
(60 kg + 40 kg) * final speed after the collision = 300 kg*m/s
Simplifying the equation:
100 kg * final speed after the collision = 300 kg*m/s
Dividing both sides of the equation by 100 kg:
final speed after the collision = 300 kg*m/s / 100 kg
= 3 m/s
Therefore, the final speed after the collision is 3 m/s.
To determine how much kinetic energy was lost in the collision, we need to calculate the initial kinetic energy and the final kinetic energy.
The initial kinetic energy of the boy can be found using the formula:
Initial kinetic energy of boy = 0.5 * mass of boy * (initial speed of boy)^2
= 0.5 * 60 kg * (5 m/s)^2
= 0.5 * 60 kg * 25 m^2/s^2
= 750 J (Joules)
The initial kinetic energy of the girl is zero, as she is initially at rest.
The final kinetic energy can be calculated using the formula:
Final kinetic energy = 0.5 * total mass * (final speed)^2
= 0.5 * (60 kg + 40 kg) * (3 m/s)^2
= 0.5 * 100 kg * 9 m^2/s^2
= 450 J
The difference between the initial kinetic energy and the final kinetic energy gives us the amount of kinetic energy lost in the collision:
Kinetic energy lost = Initial kinetic energy - Final kinetic energy
= 750 J - 450 J
= 300 J
Therefore, the amount of kinetic energy lost in the collision is 300 Joules.
60 kg * 5 m/s + 40 kg * 0 m/s = (60 kg + 40 kg) * v_final
300 kg*m/s = 100 kg * v_final
3 m/s = v_final
you have:
m1 = 60 kg
v1 = 5 m/s
m2 = 40 kg
v2 = 0 m/s (I presume she's not moving because there's no mention of her velocity)
so
m1v1 = (m1 + m2)V
V = m1v1 / (m1 + m2)
solve for v
once you find v, you plug it into your kinetic energy formula: K = 1/2mv^2
where you'll have Kf - Ki = your lost energy