A body of mass five kilograms fell vertically from rest. It reached the water’s surface after 2.2 seconds. Then, as it moved through the water, it descended vertically at a constant speed such that it covered 3.9 meters in 1.5 seconds. Find the magnitude of the change in momentum as a result of its impact with the water.
So here we’re thinking about some object which is initially at rest above the surface of a body of water. The object falls under gravity, hits the surface of the water, and then sinks into the water at a constant speed. We’re being asked to calculate the magnitude of the change in momentum of this object as a result of its impact with the water. Finding the answer to this question depends on the use of this equation: 𝑝 equals 𝑚𝑣. In other words, the momentum of an object is equal to its mass multiplied by its velocity.
Now we’re interested in calculating the change in momentum of this object, Δ𝑝. And since the mass of our object remains the same, this is equal to the mass 𝑚 multiplied by the change in velocity Δ𝑣. Since we’re told in the question that the mass of the object is five kilograms, we then need to calculate Δ𝑣 so that we can multiply it by 𝑚 and obtain the change in momentum, Δ𝑝. Since we’re interested in the change of momentum that occurs as a result of its impact with the water, the change in velocity that we’re interested in is the change in velocity that occurs as it enters the water.
So if we say that our object has a velocity 𝑣 one as it enters the water and it moves through the water with a velocity 𝑣 two, then Δ𝑣 that we’re interested in in this question is equal to 𝑣 two minus 𝑣 one. So in order to find the change in velocity Δ𝑣, we need to first find the velocity of the object as it moves through the water and then find the velocity of the object just as it enters the water. Finding the velocity of the object as it moves through the water, that’s 𝑣 two, is relatively straightforward in this question, since we’re told that the object descends vertically and at a constant speed. So since it moves in a constant direction at a constant speed, this means it has a constant velocity.
We’re told that the object covers 3.9 meters in 1.5 seconds. For objects traveling at a constant velocity, this velocity is given by the change in displacement of the object, Δ𝑠, divided by the change in time, Δ𝑡. Since this object moved 3.9 meters in 1.5 seconds, the change in displacement is 3.9 meters and the change in time is 1.5 seconds, which works out as a velocity of 2.6 meters per second.
Now it’s worth noting that since displacement is a vector quantity and we’ve described a downward movement of 3.9 meters as a displacement of positive 3.9, this means that we’ve implicitly decided that downward is our positive direction and upward is our negative direction. The fact that we ended up with a positive value for 𝑣 two means that its velocity at this point is downward, although we already knew that.
Now that we found 𝑣 two, we need to find 𝑣 one, the speed of the object just as it hits the water. Now, in order to figure this out, we need to remember that the object is initially falling under gravity. This means it experiences a constant acceleration equal to 𝑔, where 𝑔 takes a value of 9.8 meters per second squared. Remembering that since the acceleration is downward and we’ve defined downward as our positive direction, this value of 𝑔 is positive too. We’re told that the object falls vertically from rest and then it reaches the water’s surface after 2.2 seconds. Since we know the rate of acceleration, we can use this information to figure out how fast the object is traveling when it hits the water.
To do this, we need to recall an equation that tells us the velocity of an object that’s undergoing constant acceleration. This equation is 𝑣 equals 𝑢 plus 𝑎𝑡, where 𝑣 is the final velocity of an object, 𝑢 is the initial velocity of an object, 𝑎 is the constant acceleration, and 𝑡 is the period of time for which it accelerates. Now, we want to use this equation to figure out the velocity of the object as it reaches the water. That means the final velocity that we’re interested in calculating is 𝑣 one. The initial velocity is zero since the object starts from rest. The acceleration of the object is 9.8 meters per second squared. And the period of time for which the object accelerates is 2.2 seconds. This gives us a value for 𝑣 one of 21.56 meters per second.
So since the object is traveling with this velocity when it enters the water and then it travels at this constant velocity at every point afterwards, we know that the change in velocity that results from its impact with the water is equal to the difference between these quantities, 2.6 minus 21.56. This gives us a value of negative 18.96 meters per second. The fact that this quantity is negative signifies that its velocity in the positive direction decreases.
Now that we found the change in velocity, we can now calculate the corresponding change in momentum as a result of the object’s impact with the water. The mass of the object 𝑚 is five kilograms. And the change in velocity Δ𝑣 is negative 18.96 meters per second, which gives us a value for Δ𝑝 of negative 94.8 kilogram-meters per second. We can note though we got a negative value here, but this is only because we decided that up is negative and down is positive. If we had instead decided that upward is the positive direction, then we would’ve ended up with a positive answer.
Either way, the question only asks us for the magnitude of the change in momentum. And for this, we can ignore the negative sign. So this is our final answer. If a body of mass five kilograms falls vertically from rest and reaches the surface of some water after 2.2 seconds, then descends vertically through the water at a constant speed such that it covers 3.9 meters in 1.5 seconds, the magnitude of its change in momentum as a result of its impact with the water is 94.8 kilogram-meters per second.