A drum of mass ‘m’ kg was rolled down a ramp. At the bottom of the ramp its kinetic energy was 10 kJ and velocity was 20 m/s. With what velocity (in m/s) was it pushed down the ramp if its initial kinetic energy was 625 J?
5 m/s
Understanding Kinetic Energy and Velocity
This problem involves the concept of kinetic energy, which is the energy an object possesses due to its motion. The formula for kinetic energy (\(\text{KE}\)) is given by:
\[\text{KE} = \frac{1}{2}mv^2\]
where:
- \(m\) is the mass of the object (in kg)
- \(v\) is the velocity of the object (in m/s)
We are given the initial kinetic energy and the final kinetic energy and velocity of a drum rolling down a ramp. We need to find the initial velocity.
Calculating the Drum's Mass
We can first determine the mass of the drum using the information provided for the bottom of the ramp (the final state). At this point, the kinetic energy (\(\text{KE}_{final}\)) was 10 kJ and the velocity (\(v_{final}\)) was 20 m/s.
First, convert the final kinetic energy from kilojoules (kJ) to joules (J):
\(\text{KE}_{final} = 10 \, \text{kJ} = 10 \times 1000 \, \text{J} = 10000 \, \text{J}\)
Now, use the kinetic energy formula with the final values:
\[\text{KE}_{final} = \frac{1}{2} m v_{final}^2\]
Substitute the known values:
\[10000 \, \text{J} = \frac{1}{2} m (20 \, \text{m/s})^2\]
\[10000 = \frac{1}{2} m (400)\]
\[10000 = 200 m\]
Now, solve for the mass \(m\):
\[m = \frac{10000}{200}\]
\[m = 50 \, \text{kg}\]
The mass of the drum is 50 kg.
Finding the Initial Velocity
We are given the initial kinetic energy (\(\text{KE}_{initial}\)) was 625 J. We can now use this value and the calculated mass of the drum (50 kg) in the kinetic energy formula to find the initial velocity (\(v_{initial}\)).
\[\text{KE}_{initial} = \frac{1}{2} m v_{initial}^2\]
Substitute the known values:
\[625 \, \text{J} = \frac{1}{2} (50 \, \text{kg}) v_{initial}^2\]
\[625 = 25 v_{initial}^2\]
Now, solve for \(v_{initial}^2\):
\[v_{initial}^2 = \frac{625}{25}\]
\[v_{initial}^2 = 25\]
To find the initial velocity \(v_{initial}\), take the square root of both sides:
\[v_{initial} = \sqrt{25}\]
\[v_{initial} = 5 \, \text{m/s}\]
So, the velocity with which the drum was pushed down the ramp initially was 5 m/s.
Summary of Calculation Steps
| Step | Description | Calculation |
|---|---|---|
| 1 | Convert final KE to Joules | \(10 \, \text{kJ} = 10000 \, \text{J}\) |
| 2 | Calculate mass using final KE and velocity | \(10000 = \frac{1}{2} m (20)^2 \Rightarrow m = 50 \, \text{kg}\) |
| 3 | Use initial KE and mass to find initial velocity squared | \(625 = \frac{1}{2} (50) v_{initial}^2 \Rightarrow v_{initial}^2 = 25\) |
| 4 | Take square root to find initial velocity | \(v_{initial} = \sqrt{25} = 5 \, \text{m/s}\) |
Final Answer Determination
Based on the calculations, the initial velocity of the drum was 5 m/s.
Revision Table: Kinetic Energy Concepts
| Concept | Definition | Formula |
|---|---|---|
| Kinetic Energy | Energy possessed by an object due to its motion. | \(\text{KE} = \frac{1}{2}mv^2\) |
| Mass (m) | A measure of the amount of matter in an object. | Unit: kg |
| Velocity (v) | The rate of change of an object's position with respect to a frame of reference, and is a function of time. | Unit: m/s |
| Energy (KE) | The capacity to do work. | Unit: Joules (J) |
Additional Information: Energy Transformations on a Ramp
When an object rolls down a ramp, its energy changes forms. At the top of the ramp, it has potential energy (energy due to height). As it rolls down, this potential energy is converted into kinetic energy (energy of motion) and possibly rotational kinetic energy if it's rolling without slipping. Some energy might also be lost due to friction (converting to thermal energy). In this problem, we are given kinetic energy values directly, which simplifies the calculation focused purely on the relationship between kinetic energy, mass, and velocity.
The problem provides initial and final kinetic energies, suggesting we can directly use the kinetic energy formula without needing to consider potential energy or work done by other forces, unless the change in kinetic energy was related to potential energy change or work done. However, the way the question is phrased, giving explicit initial and final KE and velocities, implies a direct application of the KE formula to find unknown variables like mass or initial velocity using the given KE values.
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