Velocity Time Interval Calculation
The velocity ($V$) of an object fired upward is given by the equation: $V = 80 – 32t$ where $t$ is the time in seconds.
We need to find the time interval ($t$) when the velocity is between 32 m/sec and 64 m/sec. This can be written as an inequality:
$32 < V < 64$
Substitute the given velocity equation into the inequality:
$32 < 80 – 32t < 64$
Solving the Velocity Inequality
This compound inequality can be split into two separate inequalities:
- $32 < 80 – 32t$
- $80 – 32t < 64$
Inequality 1: Lower Bound
Solve for $t$ in the first inequality:
$32 < 80 – 32t$
Subtract 80 from both sides:
$32 - 80 < -32t$
$-48 < -32t$
Divide both sides by -32 and reverse the inequality sign:
$\frac{-48}{-32} > t$
$\frac{3}{2} > t$ or $t < \frac{3}{2}$
Inequality 2: Upper Bound
Solve for $t$ in the second inequality:
$80 – 32t < 64$
Subtract 80 from both sides:
$-32t < 64 - 80$
$-32t < -16$
Divide both sides by -32 and reverse the inequality sign:
$t > \frac{-16}{-32}$
$t > \frac{1}{2}$
Determining the Time Interval
Combine the results from both inequalities:
We found $t < \frac{3}{2}$ and $t > \frac{1}{2}$.
Combining these gives the interval for $t$:
$\frac{1}{2} < t < \frac{3}{2}$
Therefore, the velocity will be between 32 m/sec and 64 m/sec when the time $t$ is between 1/2 seconds and 3/2 seconds.
Important Questions from Speed Distance and Time
- Two cars, P and Q, start from a point X in India at 10 AM. Car P travels North with a speed of 25 km/h and car Q travels East with a speed of 30 km/h. Car P travels continuously but car Q stops for some time after travelling for one hour. If both the cars are at the same distance from X at 11:30 AM, for how long (in minutes) did car Q stop?
In a 500 m race, P and Q have speeds in the ratio of 3: 4. Q starts the race when P has already covered 140 m.
What is the distance between P and Q (in m) when P wins the race?A vehicle is moving at a speed of 12 m/s on a level road. It applies emergency brakes and starts to skid without rolling in a straight path. The deceleration of the vehicle is constant after braking and it comes to rest at a distance of 15 m. Assuming, $g = 10$ m/s$^2$, the coefficient of kinetic friction between the tyres and road is _________ [round off to 2 decimal places]
- The distance between Delhi and Agra is 233 km. A car P started travelling from Delhi to Agra and another car Q started from Agra to Delhi along the same road 1 hour after the car P started. The two cars crossed each other 75 minutes after the car Q started. Both cars were travelling at constant speed. The speed of car P was 10 km/hr more than the speed of car Q. How many kilometers the car Q had travelled when the cars crossed each other?
Two trains started at 7AM from the same point. The first train travelled north at a speed of 80km/h and the second train travelled south at a speed of 100 km/h. The time at which they were 540 km apart is _______________ AM.
