A car is moving along a straight road with a uniform acceleration. It passes through two points P and Q separated by a distance with velocity 30 km/ h and 40 km/h respectively. The velocity of the car midway between P and Q is
- $33.3\, km/h$
- $ 20\sqrt 2\, km/h$
- $ 25\sqrt 2\, km/h$
- $35\, km/h$
The Correct Option is D
Solution and Explanation
To solve this problem, we can use the equations of motion. We are given that a car is moving with uniform acceleration and passes through two points, P and Q, with velocities of 30 km/h and 40 km/h, respectively. We want to find the velocity of the car at the midpoint between P and Q.
Let's break down the problem step by step:
1. Given data:
- Initial velocity at point P (u) = 30 km/h
- Final velocity at point Q (v) = 40 km/h
- Distance between P and Q (s)
2. We can use the following equation of motion to relate velocity, initial velocity, acceleration, and distance:
\[v^2 = u^2 + 2as\]3. We want to find the velocity of the car (V) at the midpoint between P and Q. Let's call this point M.
4. First, let's find the acceleration (a) of the car using the data for points P and Q:
\[40^2 = 30^2 + 2a s\]Solving for 'a':
\[a = \frac{40^2 - 30^2}{2s} = \frac{1600 - 900}{2s} = \frac{700}{2s} = \frac{350}{s}\]5. Now, we can find the velocity at point M using the same equation of motion:
\[V^2 = u^2 + 2a \left(\frac{s}{2}\right)\]\[V^2 = (30^2) + 2 \left(\frac{350}{s}\right) \left(\frac{s}{2}\right)\]\[V^2 = 900 + 350\]\[V^2 = 1250\]Taking the square root of both sides:
\[V = \sqrt{1250} \approx 35.35 \text{ km/h}\]So, the velocity of the car at the midpoint between points P and Q is approximately 35.35 km/h.
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Concepts Used:
Motion in a straight line
The motion in a straight line is an object changes its position with respect to its surroundings with time, then it is called in motion. It is a change in the position of an object over time. It is nothing but linear motion.
Types of Linear Motion:
Linear motion is also known as the Rectilinear Motion which are of two types:
- Uniform linear motion with constant velocity or zero acceleration: If a body travels in a straight line by covering an equal amount of distance in an equal interval of time then it is said to have uniform motion.
- Non-Uniform linear motion with variable velocity or non-zero acceleration: Not like the uniform acceleration, the body is said to have a non-uniform motion when the velocity of a body changes by unequal amounts in equal intervals of time. The rate of change of its velocity changes at different points of time during its movement.