need help with kinematics problem

[ashwinkandel]

 i am having one confusion with physics problem. can u tell me the solution of oct/nov 2005 paper 2 , question no.4 with reasoning and if possible for 4 a (i), with picture. plz

[tmisterr]

well for number 4 (i), use the graph to find acceleration. Gradient, of a velocity time graph gives the acceleration so you have to draw the tangent to the graph at the origin to find the initial acceleration, make sure it is 90 degrees to the curve. find the gradient of the tangent to find the initial acceleration.

(ii) use integration. the graph is a parabola and from maths, we know that the general equation of a quadratic equation can be written as a(x-m)(x-n)=y where m and n are the roots of the parabola (where y equals zero) and a is a constant. so for this curve the equation is a(x-0)(x-0.6)=y. but we know one point, the peak, (0.3,8/100) i divided by 100 since we have to change it into metres. Replacing this point in equation, we get a(0.3-0)(0.3-0.6)=0.08 so a is -8/9. so now we have the equation, -8x²/9+8x/15=y after opening the bracket. now we can find the integral of the equation, since we know integral of an equation gives the area under the graph and this area (under a velocity time graph) will give the displacement. the integral gives
-8x³/27+8x²/30 and we have to solve for it between 0 and 0.6. since replacing 0 gives 0 just replace 0.6. the area after the arithmetic turns out to be 0.032m

b (i) we know force=mass*acceleration so using your value of acceleration from (i) and the value of mass of the trolley given, we can easily calculate the force but make sure all your values in S.I units, convert where necessary.
 (ii) as you can see from the graph, the gradient from the graph is getting less and less steep so acceleration is becoming smaller and smaller and since F=ma the resultant force it decreasing. It becomes 0 at time 0.3 seconds when trolley reaches maximum velocity and then the gradient starts decreasing, becoming more and more steep. The gradient is negative meaning acceleration is negative, so the force (ma) has changed direction and increases back to its initial value, but in the opposite direction

there is no need for a diagram, just try understand what i've written feel free to ask if you have any further queries