To calculate the initial velocity of a body that is uniformly retarded, we can use the equations of motion. Given the following information:
- Distance traveled (s) = 20 meters
- Time taken (t) = 10 seconds
- Final velocity (v) = 0 m/s (as it comes to rest)
We can use the second equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]
Where:
- \( s \) is the distance,
- \( u \) is the initial velocity,
- \( a \) is the acceleration (deceleration in this case),
- \( t \) is the time.
We can also relate \( u \), \( v \), and \( a \) using the first equation of motion:
\[ v = u + at \]
Since the body comes to rest, \( v = 0 \). Thus:
\[ 0 = u + at \]
From this, we can express acceleration \( a \):
\[ a = -\frac{u}{t} \]
Now, substituting \( a \) back into the second equation of motion:
\[ s = ut + \frac{1}{2} \left(-\frac{u}{t}\right) t^2 \] \[ s = ut - \frac{1}{2} ut \] \[ s = ut \left(1 - \frac{1}{2}\right) \] \[ s = \frac{1}{2} ut \]
Now we can solve for \( u \):
\[ u = \frac{2s}{t} \]
Substituting the values of \( s \) and \( t \):
\[ u = \frac{2 \times 20 \text{ m}}{10 \text{ s}} = \frac{40 \text{ m}}{10 \text{ s}} = 4 \text{ m/s} \]
Thus, the initial velocity \( u \) is \( \mathbf{4 \ m/s} \).
14 answers