Kevin has a mass of 77.5 kg and is skating with in-line skates. He sees his 21.00 kg younger brother up ahead standing on the sidewalk, with his back turned. Coming up from behind, he grabs his brother and rolls off at a speed of 2.18 m/s. Ignoring friction, find Kevin's speed just before he grabbed his brother.
Conservation of Linear Momentum:
M1V1 + M2*V2 = M1V3 + M2*V4
77.5V1 + 21*0 = 77.5*2.18 + 21*2.18
77.5V1 + 0 = 214.73
V1 = 2.77 m/s.
To solve this problem, we can use the principle of conservation of momentum, which states that the total momentum before an event is equal to the total momentum after the event. In this case, we can consider the event to be when Kevin grabs his younger brother.
Let's denote Kevin's mass as Mk = 77.5 kg, brother's mass as Mb = 21.00 kg, Kevin's initial speed as Vk (which is what we need to find), Kevin's final speed as Vf = 2.18 m/s, and brother's initial speed as Vb = 0 m/s since he is standing still.
The momentum before grabbing his brother is given by:
Momentum before = Mk * Vk + Mb * Vb
The momentum after grabbing his brother is given by:
Momentum after = (Mk + Mb) * Vf
According to the conservation of momentum, these two values must be equal, so we can set up the equation:
Mk * Vk + Mb * Vb = (Mk + Mb) * Vf
Plugging in the given values, we have:
77.5 kg * Vk + 21.00 kg * 0 m/s = (77.5 kg + 21.00 kg) * 2.18 m/s
Simplifying the equation, we have:
77.5 kg * Vk = 2.98 kg * m/s
Dividing both sides by 77.5 kg, we get:
Vk = 2.98 kg * m/s / 77.5 kg
Calculating this, we find:
Vk ≈ 0.0385 m/s
Therefore, Kevin's speed just before he grabbed his brother was approximately 0.0385 m/s.
To find Kevin's speed just before he grabbed his brother, we can use the principle of conservation of momentum. According to the principle, the total momentum before the event is equal to the total momentum after the event, assuming no external forces are acting.
Mathematically, the momentum of an object is given by the product of its mass and velocity: momentum = mass x velocity.
Before Kevin grabbed his brother, both of them were stationary, so their initial momenta were zero.
Let's assume Kevin's speed just before he grabbed his brother is v.
The mass of Kevin is given as 77.5 kg, and the mass of his younger brother is given as 21.00 kg. After Kevin grabs his brother, their combined mass will be 77.5 kg + 21.00 kg = 98.5 kg.
According to the conservation of momentum, the total initial momentum is zero, and the final momentum is the combined mass of Kevin and his brother multiplied by their final speed, which is 2.18 m/s.
So the equation is:
0 = 98.5 kg * 2.18 m/s
Now we can solve for the unknown speed (v) by rearranging the equation:
0 = 98.5 kg * 2.18 m/s
0 = 214.13 kg·m/s
Since the product of mass and velocity cannot be zero, we can conclude that Kevin's speed just before he grabbed his brother was indeed zero.
Therefore, Kevin's speed just before he grabbed his brother was 0 m/s.