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
- A car is moving on a horizontal surface in a straight line with a constant velocity of 3 m/s. A ball is thrown vertically upwards at time $t = 0$ from the top of the moving car with a velocity of 20 m/s. The acceleration due to gravity is 10 m/s$^2$.
At what value(s) of time $t$ in second(s), the ball is at a height of 15 m from the top of the moving car? 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?- An automobile travels from city A to city B and returns to city A by the same route. The speed of the vehicle during the onward and return journeys were constant at 60 km/h and 90 km/h, respectively. What is the average speed in km/h for the entire journey?
Two cars start at the same time from the same location and go in the same direction. The speed of the first car is 50 km/h and the speed of the second car is 60 km/h. The number of hours it takes for the distance between the two cars to be 20 km is ___________.
- Trucks (10 m long) and cars (5 m long) go on a single lane bridge. There must be a gap of at least 20 m after each truck and a gap of at least 15 m after each car. Trucks and cars travel at a speed of 36 km/h. If cars and trucks go alternately, what is the maximum number of vehicles that can use the bridge in one hour?
