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?
The problem asks for the time(s) when a ball, thrown vertically upwards from a moving car, reaches a specific height relative to the car.
Irrelevant Information Handling
The car moves horizontally at a constant velocity of 3 m/s. This horizontal motion does not affect the vertical motion of the ball relative to the car. Therefore, the car's velocity is irrelevant for this calculation.
Vertical Motion Kinematics
We focus solely on the ball's vertical motion. The relevant kinematic equation relating vertical displacement ($y$), initial vertical velocity ($v_0$), acceleration ($a$), and time ($t$) is:
$y = v_0 t + \frac{1}{2} a t^2$
Given values are:
- Initial vertical velocity, $v_0 = 20$ m/s (upwards)
- Acceleration due to gravity, $a = -10$ m/s$^2$ (downwards)
- Target vertical displacement (height relative to the car's top), $y = 15$ m
Calculating Time
Substitute the values into the kinematic equation:
$15 = (20) t + \frac{1}{2} (-10) t^2$
Simplify the equation:
$15 = 20t - 5t^2$
Rearrange the equation into the standard quadratic form ($At^2 + Bt + C = 0$):
$5t^2 - 20t + 15 = 0$
Divide the entire equation by 5 to simplify:
$t^2 - 4t + 3 = 0$
Factor the quadratic equation:
$ (t - 1)(t - 3) = 0 $
Solving for $t$, we get two possible values:
$ t - 1 = 0 \implies t = 1 \text{ s} $
$ t - 3 = 0 \implies t = 3 \text{ s} $
Conclusion
The ball is at a height of 15 m from the top of the moving car at times $t = 1$ second and $t = 3$ seconds. These correspond to options A and C.
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.
