Question

A truck is moving from rest with constant power P. if the displacement of the truck is proportional to t", where t is time, find n.

• A. 2
• B. 3/2
• C. 1/2
• D. 5/2

Answer 1

The Correct Option is B

To solve this problem, we need to understand the relationship between the power of the truck and its displacement over time.

1. The power P exerted by the truck is constant. Power is defined as the rate at which work is done, which can be expressed as: P = \frac{dW}{dt}, where W is work and t is time.
2. Work done can be expressed as the change in kinetic energy. Since the truck starts from rest, its initial kinetic energy is zero. If v is the velocity of the truck at time t, then: W = \frac{1}{2}mv^2, where m is the mass of the truck.
3. Using the power formula: P = \frac{d}{dt}\left(\frac{1}{2}mv^2\right) = mv\frac{dv}{dt}. Notice that v\frac{dv}{dt} is the relation between acceleration and velocity.
4. From the above equation, we can write: mv\frac{dv}{dt} = P, thus v\frac{dv}{dt} \propto P/m.
5. Since displacement s is the integral of velocity over time, consider: s = \int v \, dt. According to the given condition in the problem, s \propto t^n.
6. For a body moving under constant power: v \propto t^{\frac{n-1}{n}}. Since Power P = F \cdot v = ma \cdot v and using a = \frac{dv}{dt}, we substitute to get a proportionality relationship based on t^n.
7. The relationship between power, velocity, and displacement in terms of time is such that if s \propto t^n, then through derivations considering kinetic energy change over time, it emerges that n = \frac{3}{2}.

Therefore, the value of n is \frac{3}{2}. The correct answer is option: 3/2.