a sled slides down a frictionless hill. if the sled starts from rest at the top, its speed is 7.5 m/s.
a.) on a second run , the sled starts with a speed of 1.5 m/s at the top.. will it end with less than 9 m/s, 9 m/s, or more than 9 m/s
b.) find the speed of the sled at the bottom of the hill after the second run
Before trying to answer (a), let's do (b).
Due to the starting kinetic energy, the KE at the bottom will be increased from
(1/2)M(7.5)^2 to (1/2)M[(1.5)^2 + (7.5)^2] = (1/2)M(58.5) = (1/2)M(7.64)^2
The velocity at the bottom in the second case will be 7.64 m/s.
a.) In this scenario, the sled starts with a speed of 1.5 m/s at the top of the hill. Since there is no friction, the total mechanical energy of the sled is conserved. Therefore, the sum of the kinetic energy (K.E.) and the potential energy (P.E.) at the top of the hill should be equal to the sum of the K.E. and the P.E. at the bottom of the hill.
At the top of the hill, the sled only has potential energy (P.E.) which can be calculated using the equation:
P.E. = m * g * h
Where:
m = mass of the sled
g = acceleration due to gravity (approx. 9.8 m/s²)
h = height of the hill
Since the sled starts from rest, the initial kinetic energy (K.E.) is 0.
At the bottom of the hill, the sled has both potential energy and kinetic energy. Using the conservation of mechanical energy, we can write:
P.E. at the top = K.E. + P.E. at the bottom
m * g * h = (1/2) * m * v²
Where:
m = mass of the sled
g = acceleration due to gravity (approx. 9.8 m/s²)
h = height of the hill
v = velocity of the sled at the bottom of the hill
To determine whether the sled will end with less than 9 m/s, 9 m/s, or more than 9 m/s, we need to solve the equation for v.
m * g * h = (1/2) * m * v²
Simplifying the equation:
v² = 2 * g * h
Plugging in values:
v² = 2 * 9.8 m/s² * h
v² = 19.6 m/s² * h
v = sqrt(19.6 m/s² * h)
From this equation, we can see that the final velocity (v) at the bottom of the hill depends on the height (h) of the hill. Since h is not given, we cannot determine the exact value of v. However, we can conclude that the final velocity (v) at the bottom of the hill can be greater, less than, or equal to 9 m/s depending on the height of the hill.
b.) As mentioned earlier, we cannot determine the exact speed of the sled at the bottom of the hill after the second run without knowing the height of the hill (h).
To find the answer to these questions, we can use the principle of conservation of mechanical energy. On a frictionless hill, the only forms of energy involved are kinetic energy (KE) and potential energy (PE).
a.) According to the conservation of mechanical energy, the sum of the initial kinetic energy and potential energy at the top of the hill should be equal to the sum of the final kinetic energy and potential energy at the bottom of the hill.
We can set up the equation as follows:
Initial KE + Initial PE = Final KE + Final PE
Since the sled starts from rest in the first run, it only has potential energy at the top, which is converted into kinetic energy at the bottom:
0 + Initial PE = Final KE + 0
In the second run, the sled starts with a speed of 1.5 m/s at the top. Therefore, it has both kinetic energy and potential energy initially:
Initial KE + Initial PE = Final KE + Final PE
Now, let's compare the situation from the first run to the second run. We know that for the sled to reach a speed of 7.5 m/s at the bottom in the first run, it started from rest. So, in the second run, if the sled starts with a speed of 1.5 m/s, which is less than 7.5 m/s, it will end up with less than 9 m/s at the bottom. Therefore, the answer is the sled will end with "less than 9 m/s."
b.) To find the speed of the sled at the bottom of the hill after the second run, we need to know the height of the hill and make use of the conservation of mechanical energy principle.
When the sled starts at the top of the hill with a speed of 1.5 m/s, there is initial kinetic energy (KE) and potential energy (PE) present. At the bottom of the hill, the sled will have some amount of final kinetic energy, assuming no energy is lost due to friction.
The equation we can use is:
Initial KE + Initial PE = Final KE + Final PE
The initial KE can be calculated using the formula: KE = (1/2) * m * v^2, where m is the mass of the sled and v is its initial velocity.
Since the question doesn't provide any values for the height of the hill, the mass of the sled, or any other relevant information, it is not possible to determine the exact speed of the sled at the bottom of the hill after the second run.